Radicals and Rational Exponents

Transcription

1 Radicals and Rational Exponents CHAPTER 9 Purestock/SuperStock Math at Work: Computer Support Specialist OUTLINE Study Strategies: Know Your School 9.1 Radical Expressions and Functions 9.2 Rational Exponents 9. Simplifying Expressions Containing Square Roots 9. Simplifying Expressions Containing Higher Roots 9. Adding, Subtracting, and Multiplying Radicals 9.6 Dividing Radicals Putting It All Together 9.7 Solving Radical Equations 9.8 Complex Numbers Group Activity empowerme: My School As a computer support specialist, Pilar Benitez is confident in her technical and communication skills. She knows that she has to maintain a firm grasp on everything her company does and understand how it fits with changes in the larger tech industry. And, she has to be able to explain all of this to customers. Our customers are small-business people and individuals, Pilar describes. When they have a problem with one of our products, I m the person who helps them resolve it. But starting a new job at a large American software company requires that Pilar learn about how the company operates before she can do her best work. What products do they offer customers? Where can she find the technical specifications for all of the products? How are she and the rest of her team expected to share responsibilities? If she cannot answer a question, where can she find an answer? These are just some of the questions that need answers. Pilar knows that if she can learn about her new company s policies and procedures, she will be off to a successful start. The same is true for students attending a new school; the more students know about their school s operation and culture, the more tools they have to succeed. In this chapter, we will learn about working with radicals and rational exponents and offer strategies for learning about your school. 27

2 Study Strategies Know Your School Understanding how your college or university works and where to find resources are critical components of being successful in college. Where do you go if you have a question about financial aid? Where can you find out about clubs that are of interest to you? Where do you go if you have questions about which classes you still need to take in order to graduate? Let s use the P.O.W.E.R. framework to help you get to know your school. I will learn about the different offices, resources, and services at my school as well as how my school works. Complete the empowerme survey that appears before the Chapter Summary to learn how well you know your school. Think about what you know and what you still need to learn. Know your school s website. Identify important places on the website such as where you can register for classes, pay your tuition online, how to make an appointment to see an adviser, and where you can find information about the tutoring center. Identify places you need to know at your school: important offices where you can get questions answered, services offered at your school, organizations or clubs that you can join, etc. Identify important deadlines you should know such as registering for classes, paying tuition, and submitting financial aid applications. Identify academic services that you may want to use such as the library, a tutoring center, and a testing center. Identify support services that may be helpful to you such as the campus health center, a veterans support office, an office to help students with disabilities, and a child care center. Be sure you have your syllabus for each class with information about the class as well as the location of your instructors offices and their office hours. It s time to learn about some of the offices, resources, and support services at your school and to learn about some of the procedures. Some important, common offices and resources are listed here, but different schools have different names for these services. Some apply to all students, and some may not apply to you. For the items that apply to you, fill in the blanks with the information for your school. My school s website is. Among other things, know how you can register for classes and pay your tuition online. The office where I can go to register for classes is called the office. Its location is. Know the deadlines for registering for classes. When you are choosing your classes, know where they are located. If the campus is large, be sure you have enough time to get from one class to the next. The office where I can go to ask questions about my tuition bill or to pay my bill in person is called the office. Its location is. Know when tuition is due. The office where I can go to ask questions about financial aid, get financial aid forms, or turn in forms is called the office. Its location is. Know when to get the financial aid process started and when the forms are due. The office where I can talk with someone about choosing a major or get help choosing classes is called the office. Its location is. 28 CHAPTER 9 Radicals and Rational Exponents

3 The office where I can go to talk with someone about personal problems that are interfering with my success in college is called the office. Its location is. Does your school have an emergency alert system? If so, know how to sign up for it. You may have options of receiving an , text, or automated phone call. Most institutions have a Tutoring Center or a Math Lab that is available to students free of charge. The place I can go to get free help with my math class is called the. Its location is, and its hours are. What are the procedures for seeing a tutor? My math instructor s office is located in, and his/her office hours are. Most schools have a testing center where students can go if they are allowed extra time on their tests or if they need to take a make-up test. At my school, this center is called the. Its hours are. What are the procedures for taking a test in their center? The office where I can learn about the resources available to me as a student with a disability is called the office. Its location is. Most campuses have an office that helps veterans. On my campus, that office is called the. Its location is. Many campuses have a child care center for students with children. I can get information about the child care center at. The place I can go to find out about campus clubs, events, and organizations is called the. Most institutions have a place where students can get help finding a job or preparing for an interview. At my school, this office is called the office. Its location is. Do you feel like you have a good understanding of how your school works and where to go for help? Did you learn the names and locations of the offices listed in the Work section? If you feel like you learned about the places and services that will be beneficial to you, be sure to use them when you need them! Also, guide classmates to the proper place when they need help. If you did not learn everything you think you need to know, ask one of your instructors or go to an information office at your school. Search the institution s website. Are there other services or resources that you feel like you need but did not learn about? Talk to one of your instructors and ask if there is such a place at your school. oliveromg/shutterstock CHAPTER 9 Radicals and Rational Exponents 29

4 Chapter 9 Plan What are your goals for Chapter 9? How can you accomplish each goal? 1 Be prepared before and during class. 2 Understand the homework to the point where you could do it without needing any help or hints. Use the P.O.W.E.R. framework to help you learn about your school. Write your own goal. Complete the empowerme that appears before the Chapter Summary to find out how much you know about your school. Read the Study Strategies to learn how to use the P.O.W.E.R. framework to learn about the resources at your school. What are your objectives for Chapter 9? 1 Be able to simplify square roots and higher roots. Be able to graph radical functions and determine the domain. 2 Be able to perform operations with rational exponents. Be able to simplify expressions containing square roots and higher roots including expressions with variables. Be able to add, subtract, and multiply radicals. Be able to divide radical expressions and rationalize denominators. How can you accomplish each objective? Learn the different definitions for square roots and higher roots. Be able to make a table of values for radical functions by choosing a proper domain. Learn the definition of a 1/n, a m/n, and a m/n. Review the rules of exponents found in Sections 6.1 and 6.2. Know how to convert from radical form to exponential form. Know the product rules and the quotient rules. Learn how to tell when a radical expression is simplified. Know how to simplify exponents with even and odd exponents. Review the procedure for Simplifying Radicals. Know the definition of like radicals. Review the rules for multiplying polynomials so you can apply them to multiplying radicals. Learn the procedure for Multiplying Radicals. Learn the procedure for Rationalizing Denominators, including denominators with two terms. Be able to find the conjugate of a radical expression, when appropriate. Know how to determine when radical expressions are simplified. 0 CHAPTER 9 Radicals and Rational Exponents

5 Chapter check your solutions. 9 6 Be able to solve radical equations, and 7 Be able to simplify, add, subtract, multiply, and divide with complex numbers. 8 Write your own goal. Plan Know the procedure for Solving Radical Equations Containing Square Roots. Be able to determine when a solution is an extraneous solution. Check all solutions. Know the definition of i. Know the procedure for Adding and Subtracting Complex Numbers. Know the procedure for Multiplying and Dividing Complex Numbers. Read Sections 9.1 to 9.8, and complete the exercises. Complete the Chapter Review and Chapter Test. How did you do? Can you simplify, add, subtract, multiply, and divide radical expressions without looking at your notes or the book? Which do you need to practice more? In your own words, explain the relationship between fractional exponents and radicals. Do you understand how to solve the different types of radical equations, or do you need more practice? Do you understand the operations with complex numbers? How are the operations similar to and different from operations with polynomials? After doing the empowerme and Study Strategy, do you know more about your school than you did before? Is there anything else you would like to know? CHAPTER 9 Radicals and Rational Exponents 1

6 9.1 Radical Expressions and Functions What are your objectives for Section 9.1? 1 Find Square Roots and Principal Square Roots 2 Find Higher Roots Evaluate n a n Determine the Domains of Square Root and Cube Root Functions Graph a Square Root Function 6 Graph a Cube Root Function How can you accomplish each objective? Write the definitions of square root, principal square root, negative square root, square root symbol or radical sign, radicand, and radical in your notes with an example next to each. Write the properties for Radicands and Square Roots in your notes, and learn them. Complete the given examples on your own. Complete You Trys 1 and 2. Know the definition of an index. Understand the definitions of the nth root, and write examples in your notes that match the definitions. Complete the given examples on your own. Complete You Trys and. Write the procedure for Evaluating n a n in your own words. Complete the given example on your own. Complete You Try. Know how to identify a radical expression, radical function, and a square root function. Write the definition of the domain of a square root function and domain of a cube root function in your own words. Complete the given examples on your own. Complete You Trys 6 8. Follow Example 9 to create your own procedure for Graphing a Square Root Function. Complete You Try 9. Follow Example 10 to create your own procedure for Graphing a Cube Root Function. Complete You Try 10. Read the explanations, follow the examples, take notes, and complete the You Trys. Recall that exponential notation represents repeated multiplication. For example, 2 means, so 2 = 9. 2 means , so 2 = 16. In this chapter we will study the opposite, or inverse, procedure, finding roots of numbers. 2 CHAPTER 9 Radicals and Rational Exponents

7 1 Find Square Roots and Principal Square Roots EXAMPLE 1 In-Class Example 1 Find all square roots of 81. Answer: 9, 9 [ YOU TRY 1 ] Find all square roots of 2. To find a square root of 2 ask yourself, What number do I square to get 2? Or, What number multiplied by itself equals 2? One number is since 2 = 2.Another number is since ( ) 2 = 2.So, and are square roots of 2. Find all square roots of 6. The 0 symbol represents the positive square root, or the principal square root, of a nonnegative number. For example, _ 2 =. BE CAREFUL Notice that _ 2 =, but _ 2. The 0 symbol represents only the principal square root (positive square root). To find the negative square root of a nonnegative number we must put a negative sign in front of the 0. For example, _ 2 =. Next we will define some terms associated with the 0 symbol. The symbol 0 is the square root symbol or the radical sign. The number under the radical sign is the radicand. The entire expression, _ 2,is called a radical. Radical sign _ 2 Radicand = 2 Radical EXAMPLE 2 In-Class Example 2 Find each square root, if possible. a) _ 16 b) _ 6 9 c) _ d) 9 e) 6 Answer: a) b) c) 6 d) _ 10 7 e) not a real number In Example 2, we are finding the principal square roots of 100 and _ and the negative 2 81 square roots of 16 and _ 9. Find each square root, if possible. a) 100 b) _ 16 c) _ 2 d) _ 81 9 e) 9 a) 100 = 10 since (10) 2 = 100. b) _ 16 means 1 _ 16. Therefore, _ 16 = 1 _ 16 = 1 =. _ c) _ 2 = 2 since ( 2 2 ) = _ d) _ 9 means 1 81 _ 9. So, _ 81 9 = 1 _ 81 9 = = 9 7. e) To find 9,ask yourself, What number do I square to get 9? There is no such real number since 2 = 9 and ( ) 2 = 9. Therefore, 9 is not a real number. SECTION 9.1 Radical Expressions and Functions

8 [ YOU TRY 2 ] Find each square root. a) 9 b) 2 1 c) _ 6 d) _ 1 6 e) 9 Let s review what we know about radicands and add a third fact. Property Radicands and Square Roots Do you think you could estimate _ 1? How could you do that? 1) If the radicand is a perfect square, the square root is a rational number. Example: _ 16 = 16 is a perfect square = _ is a perfect square. 9 2) If the radicand is a negative number, the square root is not a real number. Example: 2 is not a real number. ) If the radicand is positive and not a perfect square, then the square root is an irrational number. Example: 1 is irrational. 1 is not a perfect square. The square root of such a number is a real number that is a nonrepeating, nonterminating decimal. Sometimes, we must plot points containing radicals. For the purposes of graphing, approximating _ a radical to the nearest tenth is sufficient. A calculator with a key will give a better approximation of the radical. 2 Find Higher Roots We saw in Example 2a) that 100 = 10 since (10) 2 = 100.We can also find higher roots of numbers like a (read as the cube root of a ), a (read as the fourth root of a ), a, etc. We will look at a few roots of numbers before learning important rules. EXAMPLE In-Class Example Find each root. a) _ 27 b) _ 6 Answer: a) b) Find each root. a) 12 b) _ 2 12 (the cube root of 12) ask yourself, What number do I cube to get a) To find 12? That number is. 12 = since = 12 Finding the cube root of a number is the opposite, or inverse procedure, of cubing a number. CHAPTER 9 Radicals and Rational Exponents

9 Do you remember the powers of whole numbers? If not, review the list in Section 1.. [ YOU TRY ] b) To find _ 2 (the fifth root of 2) ask yourself, What number do I raise to the fifth power to get 2? That number is 2. _ 2 = 2 since 2 = 2 Finding the fifth root of a number and raising a number to the fifth power are opposite, or inverse, procedures. Find each root. a) _ 27 b) 8 The symbol n n a is read as the nth root of a. If a = b, then b n = a. Index n a Radicand = a We call n the index of the radical. Radical Note When finding square roots we do not write 2 a.the square root of a is written as a, and the index is understood to be 2. We know that a positive number, say 6, has a principal square root _ ( 6, or 6) and a negative square root _ ( 6,or 6). This is true for all even roots of positive numbers: square roots, fourth roots, sixth roots, and so on. For example, 81 has a principal fourth root ( _ 81,or ) and a negative fourth root ( _ 81, or ). Write examples in your notes to help you understand the different situations. Definition nth Root For any positive number a and any even index n, n the principal nth root of a is a. n the negative nth root of a is a. _ even positive = principal (positive) root even _ positive = negative root For any negative number a and any even index n, there is no real nth root of a. _ even negative = no real root For any number a and any odd index n, n there is one real nth root of a, a. _ odd any number = exactly one root BE CAREFUL The definition means that 81 cannot be because 81 is defined as the principal fourth root of 81, which must be positive. _ 81 = SECTION 9.1 Radical Expressions and Functions

10 EXAMPLE In-Class Example Find each root, if possible. a) 27 b) _ 81 c) 81 d) _ 12 e) 6 _ 6 Answer: a) b) c) not a real number d) e) 2 [ YOU TRY ] Find each root, if possible. a) _ 16 b) _ 16 c) 16 d) _ 6 e) 6 a) To find _ 16 ask yourself, What positive number do I raise to the fourth power to get 16? Since 2 = 16 and 2 is positive, _ 16 = 2. b) In part a) we found that _ 16 = 2, so _ 16 = ( _ 16 ) = 2. c) To find 16 ask yourself, What number do I raise to the fourth power to get 16? There is no such real number since 2 = 16and ( 2) = 16.Therefore, 16 has no real root. (Recall from the definition that _ even negative has no real root.) d) To find _ 6 ask yourself, What number do I cube to get 6? Since = 6and since we know that odd any number gives exactly one root, _ 6 =. e) To find 6 ask yourself, What number do I cube to get 6? Since ( ) = 6 and since we know that odd any number gives exactly one root, 6 =. Find each root, if possible. 6 a) _ 6 6 b) _ 6 c) 6 6 d) 12 e) _ 12 Evaluate n a n Earlier we said that the _ symbol represents only the positive square root of a number. For example, 9 =.It is also true that ( ) 2 = 9 =. If a variable is in the radicand and we do not know whether the variable represents a positive number, then we must use the absolute value symbol to evaluate the radical. Then we know that the result will be a positive number. For example, a 2 = a. What if the index is greater than 2? Let s look at how to find the following roots: _ ( 2) = 16 = 2 ( ) = 6 = When the index on the radical is any positive, even integer and we do not know whether the variable in the radicand represents a positive number, we must use the absolute value symbol to write the root. However, when the index is a positive, odd integer, we do not need to use the absolute value symbol. Procedure Evaluating n a n 1) If n is a positive, even integer, then 2) If n is a positive, odd integer, then n a n = a. n a n = a. 6 CHAPTER 9 Radicals and Rational Exponents

11 EXAMPLE In-Class Example Simplify. a) ( 9)2 b) r2 c) ( 2) d) 7 p 7 e) _ (z 6) f) _ (7m + ) Answer: a) 9 b) r c) 2 d) p e) z 6 f) 7m + In your notes, summarize when you need the absolute value symbol and when you do not. Simplify. a) ( 7) 2 b) k 2 c) e) _ (y 9) f ) (8p + 1) _ ( ) 7 d) n 7 a) ( 7) 2 = 7 = 7 When the index is even, use the absolute value symbol to be certain that the result is not negative. b) k 2 = k When the index is even, use the absolute value symbol to be certain _ that the result is not negative. c) ( ) = The index is odd, so the absolute value symbol is not necessary. d) 7 n 7 = n The index is odd, so the absolute value symbol is not necessary. e) _ (y 9) = y 9 Even index: use the absolute value symbol to be certain that the result is not negative. f ) (8p + 1) = 8p + 1 Odd index: the absolute value symbol is not necessary. [ YOU TRY ] Simplify. a) ( 12) 2 b) w 2 c) _ 6 e) (t + ) 6 7 f ) (h ) 7 _ ( ) d) r Determine the Domains of Square Root and Cube Root Functions An algebraic expression containing a radical is called a radical expression. When real numbers are substituted for the variable in radical expressions like x, t + 1, and p so that the expression is defined, each value that is substituted will produce only one value for the expression. Therefore, function notation can be used to represent radical expressions. Radical functions are functions of the form f (x) = n x.let s look at some square root and cube root functions. Two examples of square root functions are f (x) = x and g(r) = _ 2r 9. EXAMPLE 6 In-Class Example 6 Let f (x) = x and g(a) = _ 2a 7. Find the function values, if possible. a) f(9) b) g() c) f( ) d) g() Answer: a) 7 b) c) not a real number d) not a real number Let f (x) = x and g(r) = _ 2r 9.Find the function values, if possible. a) f(6) b) g(7) c) f( 2) d) g() a) f (6) = _ 6 = 8 b) g (7) = 2(7) 9 = 1 9 = c) f ( 2) = 2 ; not a real number d) g() = 2() 9 = _ ;not a real number [ YOU TRY 6 ] Let f (x) = x and h(t) = _ t 10.Find the function values, if possible. a) f(2) b) h(9) c) f( 11) d) h(2) SECTION 9.1 Radical Expressions and Functions 7

12 Parts c) and d) of Example 6 illustrate that when the radicand of a square root function is negative, the function is undefined. Therefore, any value that makes the radicand negative is not in the domain of a square root function. Definition The domain of a square root function consists of all of the real numbers that can be substituted for the variable so that radicand is nonnegative. To determine the domain of a square root function, set up an inequality so that the radicand 0.Solve for the variable. These are the real numbers in the domain of the function. EXAMPLE 7 In-Class Example 7 Determine the domain of each square root function. a) f (x) = x b) g(a) = _ 2a 7 Answers: a) [0, ) b) [ 7 2, ) Determine the domain of each square root function. a) f (x) = x b) g(r) = _ 2r 9 a) The radicand, x, must be greater than or equal to zero. We write that as the inequality x 0.In interval notation, we write the domain as [0, ). b) In the square root function g(r) = _ 2r 9, the radicand, 2r 9,must be nonnegative. We write this as 2r 9 0.To determine the domain of the function, solve the inequality 2r r 9 0 The value of the radicand must be 0. 2r 9 r 9 2 Solve. [ YOU TRY 7 ] Any value of r that satisfies r 9 will make the radicand greater than or equal to 2 zero. The domain of g(r) = _ 2r 9 is 9 [ 2, ). Determine the domain of each square root function. a) h(x) = x 9 b) k(t) = 7t + 2 Two examples of cube root functions are f (x) = x and h(a) = _ a.let s look at these next. EXAMPLE 8 In-Class Example 8 Let f (x) = x and h(t) = _ t 9. Find the function values, if possible. a) f(27) b) f( 10) c) h(1) d) h(1) Answers: a) b) 10 c) 2 d) Let f (x) = x and h(a) = _ a.find the function values, if possible. a) f(12) b) f( 7) c) h(10) d) h( ) a) f (12) = 12 = b) f ( 7) = _ 7 c) h (10) = 10 = d) h ( ) = = _ 8 = 2 8 CHAPTER 9 Radicals and Rational Exponents

13 [ YOU TRY 8 ] Let f (x) = x and g(c) = 2c +.Find the function values, if possible. a) f(2) b) f( 27) c) g( 2) d) g(2) Unlike square root functions, it is possible to evaluate cube root functions when the radicand is negative. Therefore, any real number may be substituted into a cube root function and the function will be defined. Definition The domain of a cube root function is the set of all real numbers. We can write this in interval notation as (, ). n In fact we can say that when n is an odd, positive number, the domain of f (x) = x is all real numbers, or (, ).This is because the odd root of any real number is, itself, a real number. Graph a Square Root Function We need to know the domain of a square root function in order to sketch its graph. EXAMPLE 9 In-Class Example 9 Graph each function. a) f(x) = x b) g(x) = x + 1 Answers: a) See graph in the solution to Example 9a. b) y (8, ) (, 6) (, 2) (0, 1) 2 g(x) = x + 1 ( 1, 0) 9 x Graph each function. a) f (x) = x b) g(x) = x + a) In Example 7 we found that the domain of f (x) = x is [0, ). When we make a table of values, we will start by letting x = 0, the smallest number in the domain, and then choose real numbers greater than 0. Usually it is easiest to choose values for x that are perfect squares so that it will be easier to plot the points. We will also plot the point (6, _ 6 )so that you can see where it lies on the graph. Connect the points with a smooth curve. f(x) = x x f(x) (0, 210) 2 y (1, 1) (9, ) (6, 6) (, 2) f (x) = x 10 x The graph reinforces the fact that this is a function. It passes the vertical line test. SECTION 9.1 Radical Expressions and Functions 9

14 b) To graph g(x) = x + we will begin by determining its domain. Solve x + 0. In your notes, write an explanation in your own words about how to graph a square root function. x + 0 The value of the radicand must be 0. x Solve. The domain of g(x) is [, ). When we make a table of values, we will start by letting x =, the smallest number in the domain, and then choose real numbers greater than. We will choose values for x so that the radicand will be a perfect square. This will make it easier to plot the points. We will also plot the point (1, _ ) so that you can see where it lies on the graph. Connect the points with a smooth curve. g(x) = x + x g(x) (, 1) 6 (, 0) 8 (0, 2) 2 y (1, ) (, ) g(x) = x + 6 x [ YOU TRY 9 ] Graph f (x) = x Since this graph represents a function, it passes the vertical line test. Graph a Cube Root Function The domain of a cube root function consists of all real numbers. Therefore, we can substitute any real number into the function and it will be defined. However, we want to choose our numbers carefully. To make the table of values, pick values in the domain so that the radicand will be a perfect cube, and choose values for the variable that will give us positive numbers, negative numbers, and zero for the value of the radicand. This will help us to graph the function correctly. EXAMPLE 10 In-Class Example 10 Graph each function. a) f (x) = x b) g(x) = _ x 2 Answers: a) See graph in the solution to Example 10a. b) y (10, 2) (, 1) (2, 0) 8 12 (1, 1) ( 6, 2) g(x) = x 2 x Graph each function. a) f (x) = x f (x) = x x f (x) b) g(x) = _ x 1 a) Make a table of values. Choose x-values that are perfect cubes. Also, remember to choose x-values that are positive, negative, and zero. Plot the points, and connect them with a smooth curve. 10 ( 1, 1) ( 8, 2) y f(x) = x (1, 1) (0, 0) (8, 2) 10 x The graph passes the vertical line test for functions. 0 CHAPTER 9 Radicals and Rational Exponents

15 b) Remember, for the table of values we want to choose values for x that will give us positive numbers, negative numbers, and zero in the radicand. First we will determine what value of x will make the radicand in g(x) = _ x 1 equal to zero. x 1 = 0 x = 1 If x = 1, the radicand equals zero. Therefore, the first value we will put in the table of values is x = 1. Then, choose a couple of numbers greater than 1 and a couple that are less than 1 so that we get positive and negative numbers in the radicand. Also, we will choose our x-values so that the radicand will be a perfect cube. Plot the points, and connect them with a smooth curve. x g(x) = _ x 1 1 _ 1 1 = 0 = 0 2 _ 2 1 = 1 = 1 9 _ 9 1 = 8 = 2 0 _ 0 1 = _ 1 = = _ 8 = 2 10 (0, 1) ( 7, 2) y (2, 1) (1, 0) (9, 2) g(x) = x 1 10 x [ YOU TRY 10 ] Graph f (x) = _ x. Since this graph represents a function, it passes the vertical line test. Using Technology We can evaluate square roots, cube roots, or even higher roots using a graphing calculator. A radical sometimes evaluates to an integer and sometimes must be approximated using a decimal. To evaluate a square root: For example, to evaluate 9 press 2 nd x 2, enter the radicand 9, and then press ) ENTER. The result is as shown on the screen on the left below. When the radicand is a perfect square such as 9, 16, or 2, then the square root evaluates to a whole number. For example, _ 16 evaluates to and _ 2 evaluates to as shown. If the radicand of a square root is not a perfect square, then the result is a decimal approximation. For example, to evaluate _ 19 press 2 nd x 2, enter the radicand 1 9, and then press ) ENTER. The result is approximately.89, rounded to four decimal places. SECTION 9.1 Radical Expressions and Functions 1

16 To evaluate a cube root: For example, to evaluate _ 27 press MATH, enter the radicand 2 7, and then press ) ENTER. The result is as shown. If the radicand is a perfect cube such as 27, then the cube root evaluates to an integer. Since 28 is not a perfect cube, the cube root evaluates to approximately.066. To evaluate radicals with an index greater than : For example, to evaluate _ 16 enter the index, press MATH, enter the radicand 1 6, and press ENTER. The result is 2. Since the fifth root of 18 evaluates to a decimal, the result is an approximation of , rounded to four decimal places as shown. Evaluate each root using a graphing calculator. If necessary, approximate to the nearest tenth. 1) _ 2 2) 216 ) _ 29 ) 2 ) _ 102 6) Using Technology We can use a graphing calculator to find the domain of a square root function or cube root function visually. The domain consists of the x-values of the points shown on the graph. We first consider the basic shape of a square root function. To graph the equation f (x) = x, press 2 nd x 2 X,T,ϴ,n ) to the right of \Y 1 =. Press ZOOM and select 6:ZStandard to graph the equation. f(x) = x The left side of the graph begins at the point (0, 0), and the right side of the graph continues up and to the right forever. The x-values of the graph consist of all x-values greater than or equal to 0. In interval notation, the domain is [0, ). The domain of any square root function can be found using a similar approach. First graph the function, and then look at the x-values of the points on the graph. The graph of a square root function will always start at a number and extend to positive or negative infinity. For example, consider the graph of the function g(x) = x as shown. The largest x-value on the graph is. The x-values g(x) = x of the graph consist of all x-values less than or equal to. In interval notation, the domain is (, ]. Next consider the basic shape of a cube root function. To graph the equation f (x) = x press MATH, select : _ (, and press X,T,ϴ,n ) to the right of \Y 1 =. Press ZOOM and select 6:ZStandard to graph the equation as shown on the graph at right. f(x) = x 2 CHAPTER 9 Radicals and Rational Exponents

17 The left side of the graph extends down and to the left forever, and the right side of the graph extends up and to the right forever. In interval notation, the domain is (, ). This is true for any cube root function, so the domain is always (, ). Determine the domain using a graphing calculator. Use interval notation in your answer. 7) f (x) = x 2 8) g(x) = x + 9) h(x) = 2 x 10) f (x) = _ x ) f (x) = x + 12) g(x) = x ANSWERS TO [YOU TRY] EXERCISES 1) 8, 8 2) a) b) 12 c) d) 1 e) not a real number ) a) b) ) a) 2 b) 2 c) not a real number d) e) ) a) 12 b) w c) d) r e) t + f ) h 6) a) b) 17 c) not a real number d) not a real number 2 7) a) [9, ) b) [ 7, ) 8) a) _ 2 b) c) 1 d) 7 9) y 10) y ( 1, 1) (, 6) (2, 2) (7, ) (, 1) (11, 2) ( 2, 0) f (x) = x x 8 (, 2) (, 0) (2, 1) f(x) = x 12 x ANSWERS TO TECHNOLOGY EXERCISES 1) 2) 6 ). ) 18 ) 6) 7 7) [2, ) 8) [, ) 9) (, 2] 10) [ 1, ) 11) (, ) 12) (, ) 9.1 Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective 1: Find Square Roots and Principal Square Roots Decide whether each statement is true or false. If it is false, explain why. 1) _ 121 = 11 and 11. 2) _ 81 = 9 True False; _ 121 = 11 because the symbol means principal square root. ) The square root of a negative number is a negative number. False; the square root of a negative number is not a real number. ) The even root of a negative number is a negative number. False; it is not a real number. Find all square roots of each number. ) 1 12 and 12 6) and 0 7) _ 6 6 8) and and 6 Find each square root, if possible. 9) _ ) ) _ not real 12) _ 100 not real _ 81 1) _ 2 9 1) ) _ ) _ SECTION 9.1 Radical Expressions and Functions

18 Objective 2: Find Higher Roots Decide if each statement is true or false. If it is false, explain why. 17) The cube root of a negative number is a negative number. True 18) The odd root of a negative number is not a real number. False; the odd root of a negative number is a negative number. 19) Every nonnegative real number has two real, even roots. False; 20) the only even root of zero is zero. 10,000 = 10 True 21) Explain how to find _ 6. 22) Explain how to find _ 16. 2) Does 81 =? Why or why not? 2) Does _ 8 = 2? Why or why not? Find each root, if possible. 2) 12 26) 27) 29) _ ) _ 81 0) _ 1 not real 2) 1) ) 27 _ 8 2 _ not real 16 2 ) 1 1 ) ) _ 6 2 7) ) ) 6 not real 0) 16 not real _ 1) ) _ ) _ ) _ ) 9 6 6) _ 1 9 not real 7) _ 8) Objective : Evaluate n a n 9) If n is a positive, even integer and we are not certain that a 0, then we must use the absolute value symbol to evaluate n a n n. That is, a n = a. Why must we use the absolute value symbol? 0) If n is a positive, odd integer then n a n = a for any value of a. Why don t we need to use the absolute value symbol? Simplify. _ 1) _ 2) 2 2 ) ( 6) 6 _ ) ( 11) 2 11 ) y 2 y 2 6) d d 7) _ 8) ( ) 9) z 7 z 60) t 7 t _ 61) h 6 h 62) m 6 m _ 6) (x + 7) 2 x + 7 _ 6) (a 9) 2 a 9 6) (2t 1) 2t 1 66) (6r + 7) 6r ) (n + 2) n + 2 _ 68) (x 6) x 6 _ 69) 7 (d 8) 7 6 d 8 70) (y + ) 6 y + Objective : Determine the Domains of Square Root and Cube Root Functions 71) Is 1 in the domain of f (x) = x? Explain your answer. No, because 1 is not a real number. 72) Is 1 in the domain of f (x) = x? Explain your answer. Yes, because _ 1 = 1. 7) How do you find the domain of a square root function? 7) What is the domain of a cube root function? (, ) Let f (x) = x and g(t) = t +. Find each of the following, if possible, and simplify. 7) f(100) 10 76) f(9) 77) f( 9) not a real number 78) f( 6) not a real number 79) g( 1) 1 80) g() _ 1 81) g( ) not a real number 82) g( 2) not a real number 8) f(a) a 8) g(w) _ w + 8) f(t + ) _ t + 87) g(2n 1) 6n + 1 Let f (a) = following, and simplify. 86) f(p 9) _ p 9 88) g(m + 10) m + a and g(x) = x 1. Find each of the 89) f(6) 90) f( 12) 91) f( 27) 92) g() _ 11 9) g( ) _ 17 9) g(0) 1 9) g(r) r 1 96) f(z) _ z 97) f(c + 8) _ c ) f(k 2) k 2 99) g(2a ) _ 8a 1 100) g(6 w) _ 2 w Determine the domain of each function. 101) h(n) = n + 2 [ 2, ) 102) g(c) = c ) p(a) = a 8 [8, ) 10) f (a) = _ [ 10, ) a 1 [1, ) 10) f (a) = _ a 7 (, ) 106) h(t) = _ t (, ) CHAPTER 9 Radicals and Rational Exponents

19 107) r(k) = _ k ) k(x) = _ 2x 109) g(x) = 2x (, ) 111) g(t) = _ t (, 0] 11) r(a) = _ 110) h(c) = _ c 112) h(x) = x (, ] 9 7a (, 9 7] 11) g(c) = _ 8 c (, 8 ] Objective : Graph a Square Root Function Determine the domain, and then graph each function. 11) f (x) = _ x 1 116) g(x) = x 117) g(x) = x + 118) h(x) = _ x ) h(x) = x 2 121) f (x) = _ x [ 7 [, ) 2, ) 120) f (x) = x ) g(x) = _ x Objective 6: Graph a Cube Root Function Determine the domain, and then graph each function. 12) f (x) = _ x ) g(x) = _ x ) h(x) = _ x 2 126) g(x) = _ x 127) g(x) = x ) h(x) = x 1 (, ) 129) A stackable storage cube has a storage capacity of 8 ft. The length of a side of the cube, s, is given by s = V, where V is the volume of the cube. How long is each side of the cube? 2 ft 10) A die (singular of dice) has a surface area of 1. cm. Since the die is in the shape of a cube the length of a side of the die, A s, is given by s =, where A is 6 McGraw-Hill Education/Mark the surface area of the cube. How Dierker, photographer long is each side of the cube? 1. cm 11) If an object is dropped from the top _ of a building 160 h 160 ft tall, then the formula t = _ 16 describes how many seconds, t, it takes for the object to reach a height of h ft above the ground. If a piece of ice falls off the top of this building, how long would it take to reach the ground? Give an exact a nswer and an approximation to two decimal places. _ 10 sec;.16 sec 12) A circular flower garden has an area of 1 ft 2. The radius, r, of a circle in terms of its area is given by A r =, where A is the area of the circle. What is π the radius of the garden? Give an exact answer _ and an approximation to two decimal places. _ 1 ft;.0 ft π 1) The speed limit on a street in a residential neighborhood is 2 mph. A car involved in an accident on this road left skid marks 0 ft long. Accident investigators determine that the speed of the car can be described by the function S(d) = 22.d, where S(d ) is the speed of the car, in miles per hour, at the time of impact and d is the length of the skid marks, in feet. Was the driver speeding at the time of the accident? Show all work to support your answer. Yes. The car was traveling at 0 mph. 1) A car involved in an accident on a wet highway leaves skid marks 170 ft long. Accident investigators determine that the speed of the car, S(d ) in miles per hour, can be described by the function S(d) = _ 19.8d, where d is the length of the skid marks, in feet. The speed limit on the highway is 6 mph. Was the car speeding when the accident occurred? Show all work to support your answer. No. The car was traveling about 8 mph. Use the following information for Exercises The period of a pendulum is the time it takes for the pendulum to make one complete swing back and forth. The period, T(L) in _ seconds, can be described by the function L T(L) = 2π _, where L is the length of the pendulum, 2 in feet. For each exercise give an exact answer and an answer rounded to two decimal places. Use.1 for π. 1) Find the period of the pendulum whose length is 8 ft. π sec;.1 sec 16) Find the period of the pendulum whose length is π 2 ft. sec; 1.7 sec 2 17) Find T ( 1 2 ), and explain what it means in the context of the problem. 18) Find T(1.), and explain what it means in the context of the problem. 19) Find the period of a 0-inch-long pendulum. 10) Find the period of a -inch-long pendulum. R1) How well do you remember the powers of whole numbers? Refer to the list in Section 1. if you need to review them. R2) Explain, in your own words, how to determine the domain of a square root function. SECTION 9.1 Radical Expressions and Functions

20 9.2 Rational Exponents What are your objectives for Section 9.2? How can you accomplish each objective? 1 Evaluate Expressions of the Form a 1/n Write the definition of a1/n in your own words. Complete the given example on your own. Complete You Try 1. 2 Evaluate Expressions of the Form a m/n Write the definition of am/n in your own words. Complete the given example on your own. Complete You Try 2. Evaluate Expression of the Form a m/n Write the definition of a m/n in your own words. Complete the given example on your own. Complete You Try. Combine the Rules of Exponents Convert a Radical Expression to Exponential Form and Simplify Review the rules of exponents found in Sections 6.1 and 6.2 to help follow the examples. Complete the given examples on your own. Complete You Trys and. Use the same rules of exponents to convert between radical and exponential forms. Complete the given examples on your own. Complete You Trys 6 and 7. Read the explanations, follow the examples, take notes, and complete the You Trys. 1 Evaluate Expressions of the Form a1/n In this section, we will explain the relationship between radicals and rational exponents (fractional exponents). Sometimes, converting between these two forms makes it easier to simplify expressions. Definition If n is a positive integer greater than 1 and n a is a real number, then a 1/n n = a (The denominator of the fractional exponent is the index of the radical.) 6 CHAPTER 9 Radicals and Rational Exponents

21 EXAMPLE 1 In-Class Example 1 Write in radical form, and evaluate. a) 91/2 b) 271/ c) 21/ d) ( 1) 1/2 e) 16 1/ f) ( 2) 1/ Answer: a) b) c) 2 d) not a real number e) 2 f) 2 [ YOU TRY 1 ] Write in radical form, and evaluate. a) 8 1/ b) 9 1/2 c) 81 1/ d) 6 1/6 e) ( 16) 1/ f) ( 12) 1/ a) The denominator of the fractional exponent is the index of the radical. Therefore, 8 1/ = 8 = 2. b) The denominator in the exponent of 9 1/2 is 2, so the index on the radical is 2, meaning square root. 9 1/2 = _ 9 = 7 c) 81 1/ = _ 81 = d) 6 1/6 = (6 1/6 ) = 6 _ 6 = 2 e) ( 16) 1/ = 16, which is not a real number. Remember, the even root of a negative number is not a real number. f ) ( 12) 1/ = _ 12 = Write in radical form, and evaluate. The odd root of a negative number is a negative number. a) 16 1/ b) 121 1/2 c) / d) 81 1/ e) ( 9) 1/2 f) ( 12) 1/ 2 Evaluate Expressions of the Form am/n We can add another relationship between rational exponents and radicals. Definition If m and n are positive integers and m is in lowest terms, then n a m/n = (a 1/n ) m = ( n a )m provided that a 1/n is a real number. (The denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression.) We can also think of a m/n this way: a m/n = (a m ) 1/n = n _ a m. EXAMPLE 2 In-Class Example 2 Write in radical form, and evaluate. a) 8 2/ b) 16 /2 c) ( 16) /2 d) 2 / e) ( 12) 2/ Answer: a) b) 6 c) not a real number d) 16 e) 2 Write in radical form, and evaluate. a) 2 /2 b) 6 2/ c) ( 81) /2 d) 81 /2 e) ( 1000) 2/ a) The denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression. 2 /2 = (2 1/2 ) Use the definition to rewrite the exponent. = _ ( 2 ) Rewrite as a radical. = _ 2 = = 12 SECTION 9.2 Rational Exponents 7

22 b) To evaluate 6 2/, first evaluate 6 2/, then take the negative of that result. In your notes, write an explanation in your own words about how to convert from exponential form to radical form. Include examples. [ YOU TRY 2 ] 6 2/ = (6 2/ ) = (6 1/ ) 2 Use the definition to rewrite the exponent. = ( _ 6 ) 2 Rewrite as a radical. = () 2 _ 6 = = 16 c) ( 81) /2 = [( 81) 1/2 ] = _ ( 81 ) Not a real number. d) 81 /2 = (81 1/2 ) = _ ( 81 ) = (9) = 729 e) ( 1000) 2/ = [( 1000) 1/ ] 2 = ( Write in radical form, and evaluate ) 2 = ( 10) 2 = 100 The even root of a negative number is not a real number. a) 2 2/ b) 100 /2 c) ( 100) /2 d) ( 1) / e) 1 / BE CAREFUL In Example 2, notice how the parentheses affect how we evaluate an expression. The base of the expression ( 81) /2 is 81, while the base of 81 /2 is 81. Evaluate Expressions of the Form a m/n Recall that if n is any integer and a 0, then a n = ( 1 a ) = 1 a n. That is, to rewrite the expression with a positive exponent, take the reciprocal of the base. For example, We can extend this idea to rational exponents. 2 = ( ) = 16 n Definition If a m/n is a nonzero real number, then a m/n = ( 1 a m/n 1 ) = a m/n (To rewrite the expression with a positive exponent, take the reciprocal of the base.) EXAMPLE Rewrite with a positive exponent, and evaluate. In-Class Example Rewrite with a positive exponent, and evaluate. a) 9 /2 b) 81 / c) ( _ 8 27) / 1 Answer: a) 27 b) 1 27 c) a) 6 1/2 b) 2 2/ c) ( / ) 8 CHAPTER 9 Radicals and Rational Exponents

23 a) To write 6 1/2 with a positive exponent, take the reciprocal of the base. Write out the steps as you are reading the example! The negative exponent does not make the expression negative! [ YOU TRY ] 6 1/2 = ( _ 1 6 ) _ = _ 1 6 = 1 6 1/2 2/ b) 2 2/ = ( _ 1 2 ) _ = ( 2 _ 1 2 ) = ( 1 2 ) = 1 c) ( / ) = ( 6 12 ) = ( = ( 2 ) = 16 _ 2 2/ ) The reciprocal of 6 is 1 _ 6. The denominator of the fractional exponent is the index of the radical. The reciprocal of 2 is 1 _ 2. The denominator of the fractional exponent is the index of the radical. _ 1 2 = _ 1 2 The reciprocal of 12 is The denominator of the fractional exponent is the index of the radical = Rewrite with a positive exponent, and evaluate. a) 1 1/2 b) 16 / c) ( _ / ) Combine the Rules of Exponents We can combine the rules presented in this section with the rules of exponents we learned in Chapter 6 to simplify expressions containing numbers or variables. EXAMPLE In-Class Example Simplify completely. The answer should contain only positive exponents. a) (2/ ) 6 b) 272/ 27 1/ 1/ c) 7/ Answer: a) 12/ b) c) 16 Simplify completely. The answer should contain only positive exponents. a) (6 1/ ) 2 b) 2 / 2 1/ c) 82/9 8 11/9 a) (6 1/ ) 2 = 6 2/ Multiply exponents. _ + ( _ 1 b) 2 / 2 1/ = 2 ) Add exponents. = 2 2/ = 2 1/2 = SECTION 9.2 Rational Exponents 9

24 c) 82/9 2_ 8 11/9 = Subtract exponents. = 8 9/9 = = 1 ( 8 ) = 1 8 Subtract 2 9 _ Reduce 9 9. [ YOU TRY ] Simplify completely. The answer should contain only positive exponents. a) (16 1/12 ) b) 9 /8 9 1/8 c) 27/9 _ 27 16/9 EXAMPLE In-Class Example Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. a) x 1/ x / p 1/6 8 b) _ ( q /2 ) c) h 2/ h / h / m 1/2 n /2 d) ( m / n ) Answer: a) x /2 p / b) _ q 12 1 c) _ h 7/ d) m /20 n Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. a) r 1/8 r /8 b) _ x2/ ( y 1/ ) 6 c) n /6 n 1/ n 1/6 a) r 1/8 r /8 1 = r Add exponents. = r /8 = r 1/2 b) _ x2/ ( y 1/ ) 2 6= x _ 6 1 y 6 = _ x y / c) n /6 n 1/ = n _ n 6 = _ n 1/6 n 1/6 n 1/6 = n /6 n 1/6 = n 6 ( 1 6 ) = n 2/6 = n 1/ = _ 1 n 1/ d) a 7 b 1/2 / ( a b 1/ = a b 1/ / ) ( a 7 b 1/2 ) Multiply exponents. Reduce. Add exponents. Subtract exponents. d) a 7 b 1/2 / ( a c 1/ ) Eliminate the negative from the outermost exponent by taking the reciprocal of the base. Simplify the expression inside the parentheses by subtracting the exponents. = (a ( 7) b 1/ 1/2 ) / = (a + 7 b 2/6 /6 ) / = (a 12 b 1/6 ) / Apply the power rule, and simplify. [ YOU TRY ] Simplify completely. Assume the variables represent positive real numbers. The answer should contain only positive exponents. a) (a b 1/ ) 10 b) _ t/10 t 7/10 = (a 12 ) / (b 1/6 ) / = a 9 b 1/8 = a9 _ b 1/8 c) s / s 1/2 s / d) x y /8 2/ _ ( x 9 y 1/ ) 0 CHAPTER 9 Radicals and Rational Exponents

25 Convert a Radical Expression to Exponential Form and Simplify Some radicals can be simplified by first putting them into rational exponent form and then converting them back to radicals. EXAMPLE 6 In-Class Example 6 Follow the instructions for Example 6. _ 10 a) 6 8 b) x 6 Answer: a) 6 b) x In your notes, write an explanation in your own words that describes how to convert from radical form to exponential form. Include examples. [ YOU TRY 6 ] Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume the variable represents a nonnegative real number. a) 8 9 b) 6 s a) Because the index of the radical is the denominator of the exponent and the power is the numerator, we can write 8 9 = 9 /8 = 9 1/2 = Write with a rational exponent. b) 6 s = s /6 = s 2/ = s 2 Write with a rational exponent. The expression 6 s is not in simplest form because the and the 6 contain a common factor of 2, but s 2 is in simplest form because 2 and do not have any common factors besides 1. Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Assume the variable represents a nonnegative real number. a) b) 10 p In Section 9.1 we said that if a is negative and n is a positive, even number, then n a n = a. For example, if we are not told that k is positive, then k 2 = k. However, if we assume that k is positive, then k 2 = k. In the rest of this chapter, we will assume that all variables represent positive, real numbers unless otherwise stated. When we make this assumption, we do not need to use absolute values when simplifying even roots. And if we consider this together with the relationship between radicals and rational exponents we have another way to explain why n a n = a. EXAMPLE 7 In-Class Example 7 Simplify. _ a) 11 b) ( _ ) c) p 2 Answer: a) 11 b) c) p Simplify. a) b) ( 9 ) c) k 2 a) = ( ) 1/ = 1_ = 1 = b) ( 9 ) = (9 1/ ) 1_ = 9 = 9 1 = 9 _ c) k 2 = (k 2 ) 1/2 = k 2 1_ 2 = k 1 = k [ YOU TRY 7 ] Simplify. a) ( _ 10 ) 2 b) 7 c) t SECTION 9.2 Rational Exponents 1

26 Using Technology We can evaluate square roots, cube roots, or even higher roots by first rewriting the radical in exponential form and then using a graphing calculator. For example, to evaluate _ 9, first rewrite the radical as 9 1/2, then enter 9, press ^ (, enter 1 2, and press ) ENTER. The result is 7, as shown on the screen on the left below. To approximate _ 12 2 rounded to the nearest tenth, first rewrite the radical as 12 2/, then enter 1 2, press ^ (, enter 2, and press ) ENTER. The result is as shown on the screen on the right below. The result rounded to the nearest tenth is then.2. To evaluate radicals with an index greater than, follow the same procedure explained above. Evaluate by rewriting in exponential form if necessary and then using a graphing calculator. If necessary, approximate to the nearest tenth. 1) 16 1/2 2) 12 ) _ 7 ) 61 1/2 ) 096 2/ 6) 201 1/ ANSWERS TO [YOU TRY] EXERCISES 1) a) 2 b) 11 c) 10 d) e) not a real number f) 2) a) b) 1000 c) not a real number d) 1 e) 1 ) a) _ 1 12 b) 1 8 c) 9 ) a) 2 b) 7 c) _ 1 81 ) a) a 0 b 2 b) _ 1 t 2/ c) x 2 s/2 d) 6) a) b) p 2 7) a) 10 b) 7 c) t y 1/20 ANSWERS TO TECHNOLOGY EXERCISES 1) 2) 8 ) 6.1 ) 19 ) 26 6) Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective 1: Evaluate Expressions of the Form a 1/n 1) Explain how to write 2 1/2 in radical form. 2) Explain how to write 1 1/ in radical form. Write in radical form, and evaluate. ) 9 1/2 ) 6 1/2 8 ) / 10 6) 27 1/ 7) 2 1/ 2 8) 81 1/ 9) 6 1/6 2 10) 12 1/ 11) ( 1/2 121 ) 1) ( 12 1/ 6 ) _ ) ( 1/2 9 ) 1/ 1) ( _ ) 1) ( 6 1/2 169 ) _ 6 16) ( _ ) 2 2 1/ 10 _ 17) ( 81) 1/ not a real 18) ( 169) 1/2 not a real number number 19) ( 1) 1/7 1 20) ( 8) 1/ 2 2 CHAPTER 9 Radicals and Rational Exponents

27 Objective 2: Evaluate Expressions of the Form a m/n 21) Explain how to write 16 / in radical form. 22) Explain how to write 100 /2 in radical form. Write in radical form, and evaluate. 2) 8 / 16 2) 81 / 27 2) 6 /6 2 26) 2 / 8 27) ( 12) 2/ 2 28) ( 1000) 2/ ) 6 / ) 27 / 81 1) ( 81) / not a real 2) ( 2) /2 not a real number number ) ( _ / ) ) ( 1000 _ 27 ) 8 _ 27 2/ 100 ) 16 / 2 9 6) ( _ 8 27 ) / 16 _ 81 Objective : Evaluate Expressions of the Form a m/n Decide whether each statement is true or false. Explain your answer. 7) 81 1/2 = 9 8) ( /2 ) = ( / ) Rewrite with a positive exponent, and evaluate. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 9) 6 1/2 1 = ( 6 1/2 ) The reciprocal of 6 is _ ) 9 1/2 7 2) 100 1/2 ) / _ 1 ) 27 1/ 10 ) ( 1 _ 81 ) = 1 6 = 1 8 Simplify. 1 0) ( _ / ) = (1000 ) 1/ 1 The reciprocal of _ 1000 is = The denominator of the fractional 1000 exponent is the index of the radical. = 10 Simplify. 1/ 6) ( 1 _ 2 ) _ / 2 7) ( _ 1 6 1/ ) 8) ( 1 12 ) 9) 6 /6 _ 1 2 0) 81 / The denominator of the fractional exponent is the index of the radical. 1 1/ 1 _ 27 1) 12 2/ _ 1 2) 6 2/ 2 _ 1 16 ) ( 2 _ ) ) ( 6 12 ) /2 ) ( ) _ 16 ) /2 / Objective : Combine the Rules of Exponents Simplify completely. The answer should contain only positive exponents. 7) 2 2/ 2 7/ 8 8) / / 2 9) (9 1/ ) 2 60) (7 2/ ) 9 61) 8 7/ 8 / 8 / 62) 6 / 6 / 6 1/ 6) 22/ 2 / 2 2/ 6) _ 6/ / 6) _ /2 9/ _ 1 66) 6 1 7/ 6 1/2 6 /2 6 Simplify completely. The answer should contain only positive exponents. 67) z 1/6 z /6 z 68) h 1/6 h / 1 h 7/12 69) ( 9v /8 )(8v / ) 72v 11/8 70) ( x 1/ )(8x /9 ) 71) _ a/9 a /9 a1/9 72) _ x1/6 x /6 1 _ x 2/ 7) 8w/10 10w 2/ _ 2 7) 20c 2/ w 1/10 72c /6 7) (x 2/9 ) _ 1 x 2/ 76) (n 2/7 ) 77) (z 1/ ) 2/ z 2/1 78) (r / ) /2 r 10/ 79) (81u 8/ v ) / 27u 2 v 80) (6x 6 y 12/ ) /6 2x y 2 81) (2r 1/ s /9 ) / 8r 1/ s /1 82) (12a 9 b 1/ ) 2/ 2a 6 b 1/6 8) f 6/7 1/ ( 27g / ) _ f 2/7 g /9 8) 16c 8 / _ ( b 11/ ) 8) x / 6 ( w /2 x ) 10 w 9 86) t /2 ( u 1/ ) 87) y1/2 y 1/ y / y 2/ 89) a b 2/ _ ( 2a 2 b ) 91) r/ t 2 /2 _ ( r 2/ t ) 9) h 2 k /2 /6 ( h 8 k /6 ) 2/ 2 6) ( 81 _ 16 a 12/ b 2/ t 21/2 _ r 1/ 1 h k 2/18 t 6 u 88) _ t t 1/2 t / t1/ 1000 _ 27 8 _ 27 _ 18c /2 _ 1 n 6/7 90) 16c 8 d /2 _ ( c d ) 92) x10 y 1/6 2/ ( x 8 y 2/ ) 2x 1/9 8b11/ _ c 6 6 c 18 d y1/ _ x 12 9) c1/8 d 8/ ( c / d cd 8 ) 9) p 1/2 (p 2/ + p 1/2 ) p 7/6 + p 96) w / (w 1/2 w ) w 11/6 w 1/ SECTION 9.2 Rational Exponents

28 Objective : Convert a Radical Expression to Exponential Form and Simplify Rewrite each radical in exponential form, then simplify. Write the answer in simplest (or radical) form. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. _ 12 97) 2 6 = 2 6/12 Write with a rational exponent. = 2 1/2 Reduce the exponent. = Evaluate ) c = c /10 Write with a rational exponent. = c 2/ Reduce the exponent. = Write in radical form. c2 _ 6 99) ) ) ) 2 10) ( ) 2 10) ( _ 10 ) 10 10) ( _ 12 ) ) ( _ 1 ) 1 _ 107) x 12 x 108) 8 t t ) k 2 _ 9 k 110) w 6 _ w 2 _ 111) z 2 _ 8 z 112) m m 2 11) d d 11) s 6 s The wind chill temperature, WC, measures how cold it feels outside (for temperatures under 0 degrees F) when the velocity of the wind, V, is considered along with the air temperature, T. The stronger the wind at a given air temperature, the colder it feels. The formula for calculating wind chill is WC = T.7V / TV /2 where WC and T are in degrees Fahrenheit and V is in miles per hour. ( windchillglossary.shtml) Use this information for Exercises 11 and 116, and round all answers to the nearest degree. 11) Determine the wind chill when the air temperature is 20 degrees and the wind is blowing at the given speed. a) mph 1 degrees F b) 1 mph 6 degrees F 116) Determine the wind chill when the air temperature is 10 degrees and the wind is blowing at the given speed. Round your answer to the nearest degree. a) 12 mph degrees F b) 20 mph 9 degrees F Ingram Publishing/ AGE Fotostock R1) Describe how having a good understanding of the rules of exponents has helped you do these exercises. R2) What do you understand well, and what do you need to practice more? What can you do to increase your understanding? CHAPTER 9 Radicals and Rational Exponents

29 9. Simplifying Expressions Containing Square Roots What are your objectives for Section 9.? 1 Multiply Square Roots 2 Simplify the Square Root of a Whole Number Use the Quotient Rule for Square Roots Simplify Square Root Expressions Containing Variables with Even Exponents Simplify Square Root Expressions Containing Variables with Odd Exponents 6 Simplify More Square Root Expressions Containing Variables How can you accomplish each objective? Write the definition of the product rule for square roots in your own words. Complete the given example on your own. Complete You Try 1. Write the property for When Is a Square Root Simplified? Complete the given example on your own. Complete You Try 2. Write the definition of the quotient rule for square roots in your own words. Complete the given example on your own. Complete You Try. Write the property for _ a m in your notes. Complete the given example on your own. Complete You Try. Write the procedure for Simplifying a Radical Containing Variables in your own words. Complete the given examples on your own. Complete You Trys 7. Apply the rules from previous objectives, and follow the example. Complete You Try 8. Read the explanations, follow the examples, take notes, and complete the You Trys. In this section, we will introduce rules for finding the product and quotient of square roots as well as for simplifying expressions containing square roots. 1 Multiply Square Roots Let s begin with the product 9.We can find the product like this: _ 9 = 2 = 6.Also notice that 9 = _ 9 = _ 6 = 6. We obtain the same result. This leads us to the product rule for multiplying expressions containing square roots. SECTION 9. Simplifying Expressions Containing Square Roots

30 Definition Product Rule for Square Roots Let a and b be nonnegative real numbers. Then, a b = _ a b In other words, the product of two square roots equals the square root of the product. EXAMPLE 1 Multiply. a) 2 b) x In-Class Example 1 Multiply. a) 7 b) g Answer: a) 21 b) g a) 2 = _ 2 = _ 10 b) BE CAREFUL x = _ x = _ x We can multiply radicals this way only if the indices are the same. We will see later how to multiply radicals with different indices such as t. [ YOU TRY 1 ] Multiply. a) 6 2 b) _ 10 _ r Simplify the Square Root of a Whole Number Knowing how to simplify radicals is very important in the study of algebra. We begin by discussing how to simplify expressions containing square roots. How do we know when a square root is simplified? Property When Is a Square Root Simplified? An expression containing a square root is simplified when all of the following conditions are met: 1) The radicand does not contain any factors (other than 1) that are perfect squares. 2) The radicand does not contain any fractions. ) There are no radicals in the denominator of a fraction. Note: Condition 1) implies that the radical cannot contain variables with exponents greater than or equal to 2, the index of the square root. We will discuss higher roots in Section 9.. To simplify expressions containing square roots we reverse the process of multiplying. That is, we use the product rule that says _ a b = a b where a or b are perfect squares. 6 CHAPTER 9 Radicals and Rational Exponents

31 EXAMPLE 2 In-Class Example 2 Simplify completely. a) _ b) _ 72 c) _ 12 d) _ 1 Answer: a) b) 6 2 c) 2 d) _ 1 What is the first question you should ask when simplifying a square root expression? Simplify completely. a) _ 18 b) 00 c) _ 21 d) _ 8 a) The radical _ 18 is not in simplest form since 18 contains a factor (other than 1) that is a perfect square. Think of two numbers that multiply to 18 so that at least one of the numbers is a perfect square: 18 = 9 2. (While it is true that 18 = 6,neither 6 nor is a perfect square.) Rewrite _ 18 : _ 18 = is a perfect square. Product rule = 2 9 = 2 is completely simplified because 2 does not have any factors that are perfect squares. = 9 2 b) Does 00 have a factor that is a perfect square? Yes! 00 = 100.To simplify 00,rewrite it as _ 00 = is a perfect square. 10 is completely simplified because does not have any factors that are perfect squares. = 100 Product rule = = 10 c) 21 = 7 Neither nor 7 is a perfect square. 21 = 1 21 Although 1 is a perfect square, it will not help us simplify _ 21. _ 21 is in simplest form. d) There are different ways to simplify _ 8.We will look at two of them. i) Two numbers that multiply to 8 are 16 and with 16 being a perfect square. We can write _ 8 = 16 = _ 16 = ii) We can also think of 8 as 12 since is a perfect square. We can write _ 8 = 12 = _ 12 = _ 2 12 Therefore, _ 8 = _ Is _ 12 in simplest form? No, because 12 = and is a perfect square. We must continue to simplify. _ 8 = _ 2 _ 12 = 2 = 2 = 2 2 = is completely simplified because does not have any factors that are perfect squares. Example 2(d) shows that using either _ 8 = 16 or _ 8 = 12 leads us to the same result. Furthermore, this example illustrates that a radical is not always completely simplified after just one iteration of the simplification process. It is necessary to always examine the radical to determine whether or not it can be simplified more. SECTION 9. Simplifying Expressions Containing Square Roots 7

32 Note After simplifying a radical, look at the result and ask yourself, Is the radical in simplest form? If it is not, simplify again. Asking yourself this question will help you to be sure that the radical is completely simplified. [ YOU TRY 2 ] Simplify completely. a) _ 28 b) _ 7 c) _ d) _ 72 Use the Quotient Rule for Square Roots Let s simplify 6. We can say 6 9 = 6 9 = 2. It is also true that _ 6 6 = 9 _ 9 = = 2. This leads us to the quotient rule for dividing expressions containing square roots. Definition Quotient Rule for Square Roots Let a and b be nonnegative real numbers such that b 0. Then, a b = a b The square root of a quotient equals the quotient of the square roots. EXAMPLE In-Class Example Simplify _ completely. _ 8 a) _ 9 b) 2 c) 120 d) 10 _ 6 Answer: a) 2 _ 2 7 b) c) 2 d) 6 Simplify completely. _ 9 a) _ 9 b) c) 72 6 _ d) _ 81 a) Because 9 and 9 are each perfect squares, find the square root of each separately. 9 9 = _ 9 9 Quotient rule = 7 9 = and _ 9 = 7 b) Neither 200 nor 2 is a perfect square, but if we simplify 200 we get 100, which is a 2 perfect square = = Simplify CHAPTER 9 Radicals and Rational Exponents

33 In part c), which method do you prefer? Why? [ YOU TRY ] c) We can simplify 72 using two different methods. 6 i) Begin by applying the quotient rule to obtain a fraction under one radical and simplify the fraction. _ = 6 _ 6 Quotient rule = _ 12 = _ = = 2 ii) We can apply the product rule to rewrite _ 72 then simplify the fraction _ = 12 6 Product rule 6 = 1 _ 6 _ 12 Divide out the common factor. 6 1 = _ 12 = _ = Either method will produce the same result. = 2 d) The fraction _ is in simplest form, and 81 is a perfect square. Begin by applying the 81 quotient rule. _ 81 = _ 81 Quotient rule = 9 _ 81 = 9 Simplify completely. _ 100 a) 169 b) _ 27 c) 20 d) _ 11 _ 6 Simplify Square Root Expressions Containing Variables with Even Exponents Recall that a square root is not simplified if it contains any factors that are perfect squares. This means that a square root containing variables is simplified if the power on each variable is less than 2. For example, r 6 is not in simplified form. If r represents a nonnegative real number, then we can use rational exponents to simplify r 6. r 6 = (r 6 ) 1/2 = r 6 _ 2 1 = r 6/2 = r 1 Multiplying 6 is the same as dividing 6 by 2. We can generalize this result with the 2 following statement. Property _ a m If a is a nonnegative real number and m is an integer, then _ a m = a m/2 We can combine this property with the product and quotient rules to simplify radical expressions. SECTION 9. Simplifying Expressions Containing Square Roots 9

34 EXAMPLE In-Class Example Simplify completely. a) b 2 b) 100a c) 2p 8 27 d) _ g 6 Answer: a) b b) 10a 2 c) 2p 6 d) _ g Are you writing out the examples as you are reading them? [ YOU TRY ] Simplify completely. a) z 2 b) 2 9t _ c) 18b 1 d) _ 2 _ n 20 a) z 2 = z 2/2 = z 1 = z b) 9t 2 = _ 9 t 2 = 7 t 2/2 = 7t _ c) 18b 1 = 18 b 1 Product rule = _ 9 2 b 1/2 9 is a perfect square. = 2 b 7 Simplify. = b 7 2 Rewrite using the commutative property. d) We begin by using the quotient rule. 2 _ n 20 = _ 2 _ n 20 = 16 _ 2 = _ 2 n 20/2 n 10 Simplify completely. a) y 2 b) 1p 16 _ c) c 10 d) w Simplify Square Root Expressions Containing Variables with Odd Exponents How do we simplify an expression containing a square root if the power under the square root is odd? We can use the product rule for radicals and fractional exponents to help us understand how to simplify such expressions. EXAMPLE In-Class Example Simplify _ completely. a) m 9 b) t1 Answer: a) m m b) t6 _ t Simplify completely. a) x 7 b) _ c 11 a) To simplify x 7, write x 7 as the product of two factors so that the exponent of one of the factors is the largest number less than 7 that is divisible by 2 (the index of the radical). x 7 = x 6 x 1 6 is the largest number less than 7 that is divisible by 2. = x 6 x Product rule = x 6/2 x Use a fractional exponent to simplify. = x x 6 2 = _ b) To simplify c 11, write c 11 as the product of two factors so that the exponent of one of the factors is the largest number less than 11 that is divisible by 2 (the index of the radical). c 11 = c 10 c 1 10 is the largest number less than 11 that is divisible by 2. = c 10 c Product rule = c 10/2 c Use a fractional exponent to simplify. = c c 10 2 = 60 CHAPTER 9 Radicals and Rational Exponents

35 [ YOU TRY ] Simplify completely. a) m b) z 19 We used the product rule to simplify each radical in Example. During the simplification, however, we always divided an exponent by 2. This idea of division gives us another way to simplify radical expressions. Once again, let s look at the radicals and their simplified forms in Example to see how we can simplify radical expressions using division. x 7 = x x 1 = x x c 11 = c c 1 = c c Index _ Quotient of 2 7 radical 6 _ 1 Remainder Index Quotient _ of 2 11 radical 10 _ 1 Remainder Write a sample expression that follows these steps in your notes. Procedure Simplifying a Radical Containing Variables To simplify a radical expression containing variables: 1) Divide the original exponent in the radicand by the index of the radical. 2) The exponent on the variable outside of the radical will be the quotient of the division problem. ) The exponent on the variable inside of the radical will be the remainder of the division problem. EXAMPLE 6 In-Class Example 6 Simplify completely. _ a) k b) 9v c) 2m 19 Answer: a) k 2 k _ b) 7v v c) m 9 2m Do you prefer the method in Example or Example 6 for simplifying these types of expressions? [ YOU TRY 6 ] Simplify completely. _ a) 9 t b) 16b c) y 21 a) To simplify t 9, divide: _ Quotient _ 1 t 9 = t t 1 = t _ Remainder t b) 16b = _ 16 b Product rule = b 2 b 1 2 gives a quotient of 2 and a remainder of 1. = b 2 b _ c) y 21 = _ y 21 Product rule = _ 9 y 10 y 1 Product rule 21 2 gives a quotient of 10 and a remainder of 1. = _ y 10 y 9 = = y 10 y Use the commutative property to rewrite the expression. = y _ 10 y Use the product rule to write the expression with one radical. Simplify completely. _ a) c 1 b) 100v 7 _ c) 2a SECTION 9. Simplifying Expressions Containing Square Roots 61

36 If a radical contains more than one variable, apply the product or quotient rule. EXAMPLE 7 In-Class Example 7 Simplify completely. _ a) 18m 7 n 9 f b) _ 9 g 12 Answer: a) m n 2mn f f b) g 6 It is easier to simplify expressions like these when you begin by writing them as the product or quotient of separate square roots. [ YOU TRY 7 ] Simplify completely. a) 8a 1 b b) r 27 s 8 a) 8a 1 b = _ 8 a 1 b = 2 a 7 a 1 b 1 b 1 Product rule 1 2 gives a quotient = _ of 7 and 2 2 a a remainder 7 a _ of 1. b b = 2a 7 b 2 a b = 2a 7 b 2ab b) r 27 2 gives a quotient of 1 and a remainder of 1. = 2 Use the commutative property to rewrite the expression. Use the product rule to write the expression with one radical. s 8 = r 27 Quotient rule 8 s _ r 27 Product rule = _ s 8 2 = r 1 r 1 = _ s 27 2 gives a quotient of 1 and a remainder of 1. r 1 r = _ s Use the commutative property to rewrite the expression. = r 1 r s Use the product rule to write the expression with one radical. Simplify completely. _ a) 27x 10 y 9 _ b) _ 0u 1 v 20 6 Simplify More Square Root Expressions Containing Variables Next we will look at some examples of multiplying and dividing radical expressions that contain variables. Remember to always look at the result and ask yourself, Is the radical in simplest form? If it is not, simplify completely. EXAMPLE 8 In-Class Example 8 Perform the indicated operation, and simplify completely. a) _ 8b _ 2b b) x y 6xy 6r 7 c) 2r _ Answer: a) b b) x y 2 2y c) r 2 2 Perform the indicated operation, and simplify completely. a) 6t t b) 2a b 8a 2 b 20x c) x a) 6t t = 6t t Product rule 2 = 18t = 2 18 t Product rule = 9 2 t = 9 2 t = 2 t = t 2 62 CHAPTER 9 Radicals and Rational Exponents

37 In parts b) and c), think about which method you prefer and why. _ b) 2a b 8a 2 b There are two good methods for multiplying these radicals. i) Multiply the radicands to obtain one radical. _ 2a b 8a 2 b _ = 2a b 8a 2 b = 16a b 6 Is the radical in simplest form? No. = _ 16 a b 6 = a 2 a b = a 2 b a Product rule Evaluate. Product rule Multiply. Commutative property ii) Simplify each radical, then multiply. 2a b = 2 a _ b = 2 a a 8a 2 b = 8 a 2 b b = = 2 2 a b 2 b a 2ab = 2ab 2 2b _ Then, 2a b 8a 2 b = a 2ab 2ab 2 2b = a 2ab 2 ab _ 2b Commutative property _ = 2a 2 b 2 ab 2 Multiply. = 2a 2 b 2 2 b a ab 2 = 2b a = a 2 b a Multiply. Both methods give the same result. c) We can use the quotient rule first or simplify first. 20x i) = x 20x x _ = x = x = 2x 2 ii) _ 20x = 20 x x x Simplify first by using the product rule. x 2 x = _ Product rule; simplify x. x 2 x 2 x = _ = 2 x 2x _ 2 x = x Product rule = 2x 2 Divide out the common factor. Both methods give the same result. In this case, the second method was longer. Sometimes, however, this method can be more efficient. [ YOU TRY 8 ] Perform the indicated operation, and simplify completely. a) 2n _ 6n _ b) 1cd c 2 d c) 128k 9 2k SECTION 9. Simplifying Expressions Containing Square Roots 6

38 ANSWERS TO [YOU TRY] EXERCISES 1) a) _ 0 b) 10r 2) a) 2 7 b) c) _ d) 6 2 ) a) _ 10 1 b) c) 2 d) _ 11 6 ) a) y b) 12p8 c) c 6 d) _ w 2 ) a) m2 m b) z 9 z 6) a) c 6 c b) 10v v c) _ a 2a 7) a) x y _ y b) 2u _ 6 10u 8) a) 2n 2 b) cd _ c c) 8k v 10 *Additional answers can be found in the Answers to Exercises appendix. Unless otherwise stated, assume all variables represent nonnegative real numbers. Objective 1: Multiply Square Roots Multiply and simplify. 1) _ ) 11 ) _ 10 ) 6 _ 0 _ y 6y ) 7 6) 2 1 _ p p Objective 2: Simplify the Square Root of a Whole Number Label each statement as true or false. Give a reason for your answer. 7) _ 20 is in simplest form. False; 8) _ 20 contains a factor of which is a perfect square. is in simplest form. 9) _ 2 is in simplest form. 10) _ 6 is in simplest form. False; 6 contains a factor of 9 which is a perfect square. Simplify completely. 9. Exercises Do the exercises, and check your work. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 11) _ 60 = 1 Factor. = 1 Product rule = 2 1 Simplify. 12) _ = Factor. = Product rule = 10 2 Simplify. Simplify completely. If the radical is already simplified, then say so. 1) _ ) _ ) _ 6 16) _ ) _ simplified 18) _ 1 simplified 19) ) _ 80 21) _ ) _ ) _ 8 simplified 2) _ 6 simplified 2) ) ) _ ) Objective : Use the Quotient Rule for Square Roots Simplify completely. _ 1 29) 2 _ 12 0) 16 _ ) _ ) _ _ 8 11 ) ) ) _ 2 0 6) _ 2 2 7) ) 6 2 9) 7 0) _ _ 6 1) _ 9 _ 6 7 2) 2 _ _ ) _ 16 ) 60 _ 9 2 _ 1 7 Objective : Simplify Square Root Expressions Containing Variables with Even Exponents Simplify completely. ) x 8 x 6) q 6 _ q 7) w 1 w 7 8) t 16 8 t 9) 100c 2 10c _ 0) 9z 8 z 6 CHAPTER 9 Radicals and Rational Exponents

39 _ 1) 6k 6 m 10 8k m 2) 2p 20 q 1 p 10 q 7 ) 28r 2r 2 7 ) 27z 12 z 6 ) 00q 22 t 16 10q 11 t 8 6) 0n y n 2 y 2 2 7) 81 c 6 9 c 8) h _ h 1 9) 0 _ ) 18 t t m 0 _ 2 m 1 61) _ 7x 2 y 12 x y 6 62) w 2 z wz 9 Objective : Simplify Square Root Expressions Containing Variables with Odd Exponents Simplify completely. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 6) w 9 = w 8 w 1 Factor. = w 8 Product rule = w w 1 w Simplify. 6) z 19 = z _ 18 z 1 Factor. = z 18 z 1 Product rule = z 9 z Simplify. 6) a _ 67) g 1 _ 69) h 1 a 2 a g 6 g h 1 h _ 6x 2x _ q 1q t _ 71) 72x 7) 1q 7 7) 7t 11 t 77) c 8 2 d c d 79) a b a 2 b b _ 81) u v 7 u 2 v uv _ 8) 6m 9 n 6m n 2 m 8) x 12 y 2x 6 y 2 11y 66) c 7 _ 68) k 1 _ 70) b 2 c c k 7 k b 12 b 72) 100a 10a 2 a 7) 20c 9 2c c 76) p 17 p 8 p 78) r s 12 r 2 s 6 80) x 2 y 9 xy y 82) f g 9 fg fg _ 8) t 6 u 2t u 2 u _ 86) 6c 7 d c d 2 7c 87) _ 2t u 7 t 2 u 2tu 81b 6 a a 9b 89) _ a 7 _ r 91) r 9 s 2 r s 88) 12k l 9 kl kl 90) _ x 9y 6 _ x2 x 7y 92) _ 17h 11 k 8 h 17h k Objective 6: Simplify More Square Root Expressions Containing Variables Perform the indicated operation, and simplify. Assume all variables represent positive real numbers. 9) _ ) 8 6 9) _ ) 2 _ ) _ w w w 98) d d 11 d 7 _ 99) n n n n 100) a 10 a a 6 a 101) _ 2k 8k k 102) z 9 z z 6 10) a 6 b 10ab 10) 6x y 2x y 2 _ 10) 8c 9 d 2 cd 7 106) 6t u t 7 u 107) _ 18k 11 2k k 108) 8m 1 m 9 m _ 109) 120h 8 _ h 2 2h ) _ 72c 10 6c 2 2c _ 111) _ 0a 16 b 9 _ a 7 b a b 2 10ab 112) _ 21y 8 z 18 yz 1 y z 2 7yz 11) The velocity v of a moving object can be determined from its mass m and its kinetic energy KE using the 2KE formula v =, where the velocity is in meters/ m second, the mass is in kilograms, and the KE is measured in joules. A 600-kg roller coaster car is moving along a track and has kinetic energy of 120,000 joules. What is the velocity of the car? 20 m/sec 11) The length of a side s of an equilateral triangle is a function _ of its area A and can be described by s (A) = A. If an equilateral triangle has an area of 6 cm2, how long is each side of the triangle? 2 6 cm R1) Explain how you know when a square root expression is simplified and when it is not. R2) After completing the exercises, do you feel that you have learned the definitions, properties, and procedures by practicing them? Explain. R) Look at your homework and circle the types of problems that were difficult for you. Then, identify the step where you had trouble understanding what to do. What do these problems have in common? SECTION 9. Simplifying Expressions Containing Square Roots 6

40 9. Simplifying Expressions Containing Higher Roots What are your objectives for Section 9.? 1 Multiply Higher Roots 2 Simplify Higher Roots of Integers Use the Quotient Rule for Higher Roots Simplify Radicals Containing Variables Multiply and Divide Radicals with Different Indices How can you accomplish each objective? Write the product rule for higher roots in your own words. Complete the given example on your own. Complete You Try 1. Write the property for When Is a Radical Simplified? in your own words. Know the two different methods for simplifying higher roots. Complete the given example on your own. Complete You Try 2. Write the quotient rule for higher roots in your own words. Complete the given example on your own. Complete You Try. Write the property for n _ a m in your notes. Complete the given examples on your own. Complete You Trys 6. Create a procedure by following Example 7, and write it in your notes. Complete You Try 7. Read the explanations, follow the examples, take notes, and complete the You Trys. In Section 9.1 we first discussed finding higher roots like _ 16 = 2 and 27 =. In this section, we will extend what we learned about multiplying, dividing, and simplifying square roots to doing the same with higher roots. 1 Multiply Higher Roots Definition Product Rule for Higher Roots If n a and n b are real numbers, then n a n b = n a b This rule enables us to multiply and simplify radicals with any index in a way that is similar to multiplying and simplifying square roots. 66 CHAPTER 9 Radicals and Rational Exponents

41 EXAMPLE 1 In-Class Example 1 Multiply. a) 8 b) n 21 Answer: _ a) 2 b) 21n Multiply. a) 2 7 b) _ t _ 10 a) 2 7 = 2 7 = _ 1 b) _ t _ 10 = _ t 10 = _ 10t [ YOU TRY 1 ] Multiply. a) 6 b) 8 k 2 BE CAREFUL Remember that we can apply the product rule in this way only if the indices of the radicals are the same. Later in this section we will discuss how to multiply radicals with different indices. 2 Simplify Higher Roots of Integers In Section 9. we said that a simplified square root cannot contain any perfect squares. Next we list the conditions that determine when a radical with any index is in simplest form. Summarize this property in your notes. Property When Is a Radical Simplified? Let P be an expression and let n be an integer greater than 1. Then n P is completely simplified when all of the following conditions are met: 1) The radicand does not contain any factors (other than 1) that are perfect nth powers. 2) The exponents in the radicand and the index of the radical do not have any common factors (other than 1). ) The radicand does not contain any fractions. ) There are no radicals in the denominator of a fraction. Note Condition 1) implies that the radical cannot contain variables with exponents greater than or equal to n, the index of the radical. To simplify radicals with any index, use the product rule n a b = n a n b, where a or b is an nth power. Remember, to be certain that a radical is simplified completely, always look at the radical carefully and ask yourself, Is the radical in simplest form? SECTION 9. Simplifying Expressions Containing Higher Roots 67

42 EXAMPLE 2 In-Class Example 2 Simplify completely. _ a) 81 b) 1 Answer: a) b) 2 9 In your notes and in your own words, explain how you know that a radical with any index is in simplest form. [ YOU TRY 2 ] Simplify completely. a) 20 b) _ 8 a) We will look at two methods for simplifying 20. i) Since we must simplify the cube root of 20, think of two numbers that multiply to 20 so that at least one of the numbers is a perfect cube. 20 = = is a perfect cube. = 12 2 Product rule Is = 2 12 = 2 in simplest form? Yes, because 2 does not have any factors that are perfect cubes. ii) Use a factor tree to find the prime factorization of 20: 20 = 2. _ 20 = 2 2 is the prime factorization of 20. = 2 product rule = 2 = = 2 Commutative property We obtain the same result using either method. b) We will use two methods for simplifying _ 8. i) Since we must simplify the fourth root of 8, think of two numbers that multiply to 8 so that at least one of the numbers is a perfect fourth power. 8 = 16 _ 8 = _ is a perfect fourth power. = _ 16 Product rule = 2 _ Is 2 16 = 2 in simplest form? Yes, because does not have any factors that are perfect fourth powers. ii) Use a factor tree to find the prime factorization of 8: 8 = 2. 8 = 2 2 is the prime factorization of 8. = 2 Product rule = 2 2 = 2 Once again, both methods give us the same result. Simplify completely. a) _ 0 b) _ 96 Use the Quotient Rule for Higher Roots Definition Quotient Rule for Higher Roots If n a and n b are real numbers, b 0, and n is a natural number then n a n b = _ a n b 68 CHAPTER 9 Radicals and Rational Exponents

43 We apply the quotient rule when working with nth roots the same way we apply it when working with square roots. EXAMPLE In-Class Example Simplify completely. _ 8 _ a) _ b) _ 6 2 Answer: _ a) 2 b) 28 Make a list of perfect cubes and numbers raised to fourth and fifth powers. [ YOU TRY ] Simplify completely. a) _ 81 b) _ a) We can think of _ as 81 or _ 81. Let s think of it as 81. Neither 81 nor is a perfect cube, but if we simplify 81 we get 27, which is a perfect cube. _ 81 = 27 = b) Let s begin by applying the quotient rule to obtain a fraction under one radical, then simplify the fraction. 96 = _ Quotient rule = _ 8 Simplify _ = is a perfect cube. = 8 6 Product rule = = 2 Is 2 6 in simplest form? Yes, because 6 does not have any factors that are perfect cubes. Simplify completely. _ a) 160 b) _ 162 Simplify Radicals Containing Variables In Section 9.2 we discussed the relationship between radical notation and fractional exponents. Recall that Property _ n a m If a is a nonnegative number and m and n are integers such that n > 1, then n _ a m = a m/n. That is, the index of the radical becomes the denominator of the fractional exponent, and the power in the radicand becomes the numerator of the fractional exponent. This is the principle we use to simplify radicals with indices greater than 2. SECTION 9. Simplifying Expressions Containing Higher Roots 69

44 EXAMPLE In-Class Example Simplify completely. _ a) x 12 b) 8m 9 n 6 _ 6 c) _ f 12 g 2 Answer: a) x b) 2m n 2 f2 c) g [ YOU TRY ] Simplify completely. _ a) y 1 b) _ 16t 2 u 8 c) c 10 d 0 _ a) y 1 = y 1/ = y _ b) 16t 2 u 8 = 16 2 t u 8 Product rule 2/ = 2 t u 8/ Write with rational exponents. c) = 2t 6 u 2 Simplify exponents. _ c d = _ c c 0 d d = c d Quotient rule Simplify completely. a) _ p 0 b) a b 21 c) _ m 12 16n 20 To simplify a radical expression if the power in the radicand does not divide evenly by the index, we use the same methods we used in Section 9. for simplifying similar expressions with square roots. We can use the product rule or we can use the idea of quotient and remainder in a division problem. EXAMPLE In-Class Example _ Simplify d1 completely in two ways. Answer: d 7 d Be sure to do this example both ways. Which method do you prefer and why? _ Simplify x 2 completely in two ways: i) use the product rule and ii) divide the exponent by the index and use the quotient and remainder. i) Using the product _ rule: To simplify x 2, write x 2 as the product of two factors so that the exponent of one of the factors is the largest number less than 2 that is divisible by (the index). _ x 2 = x20 x 20 is the largest number less than 2 that is divisible by. _ = x20 x Product rule = x20/ x Use a fractional exponent to simplify. = x x 20 = ii) Using the quotient and remainder: _ Quotient To simplify x2, divide 2 20 _ Remainder Recall from our work with square roots in Section 9. that i) the exponent on the variable outside of the radical will be the quotient of the division problem, and ii) the exponent on the variable inside of the radical will be the remainder of the division problem. _ x 2 = x x Is x x in simplest form? Yes, because the exponent inside of the radical is less than the index, and they contain no common factors other than CHAPTER 9 Radicals and Rational Exponents

45 [ YOU TRY ] Simplify _ r 2 completely using both methods shown in Example. We can apply the product and quotient rules together with the methods in Example to simplify certain radical expressions. EXAMPLE 6 In-Class Example 6 _ Simplify 2p q 8 completely. Answer: 2pq 2 _ 2p [ YOU TRY 6 ] _ Completely simplify 6a 16 b 8. _ 6a 16 b 8 = _ 6 Simplify completely. a) _ a 16 b 8 Product rule = 8 7 a a 1 b 2 b 2 Product 16 gives a quotient 8 gives a quotient rule of and a remainder of 1. of 2 and a remainder of 2. = 2 7 a a b 2 b 2 = 2a b 2 7 a b 2 = 2a b 2 7ab 2 8x 1 y 22 b) r 19 27s 12 Simplify 8. Use the commutative property to rewrite the expression. Product rule Multiply and Divide Radicals with Different Indices The product and quotient rules for radicals apply only when the radicals have the same indices. To multiply or divide radicals with different indices, we first change the radical expressions to rational exponent form. EXAMPLE 7 In-Class Example 7 Perform the indicated operation, and write the answer in simplest radical form. a) m m b) b b _ Answer: a) m m 12 b) b Write your own procedure in your notes. [ YOU TRY 7 ] Multiply the expressions, and write the answer in simplest radical form. x 2 x The indices of x 2 and x are different, so we cannot use the product rule right now. Rewrite each radical as a fractional exponent, use the product rule for exponents, then convert the answer back to radical form. x 2 x = x 2/ x 1/2 Change radicals to fractional exponents. = x /6 x /6 Get a common denominator to add exponents. _ = x 6 + _ 6 = x 7/6 Add exponents. = 6 x 7 = x 6 x Rewrite in radical form, and simplify. Perform the indicated operation, and write the answer in simplest radical form. a) y c 2 6 y b) _ c SECTION 9. Simplifying Expressions Containing Higher Roots 71

46 2 2 ANSWERS TO [YOU TRY] EXERCISES 1) a) _ 0 b) _ 8k 2 2) a) 2 b) ) a) b) 2 ) a) p 6 b) ab 7 c) _ m 2 2n _ ) r 6 r 2 6) a) 2x y x y 2 b) r6 r 12 7) a) y 6 b) c s Mixed Exercises: Objectives 1 1) In your own words, explain the product rule for radicals. Answers may vary. 2) In your own words, explain the quotient rule for radicals. Answers may vary. ) How do you know that a radical expression containing a cube root is completely simplified? ) How do you know that a radical expression containing a fourth root is completely simplified? Assume all variables represent positive real numbers. Objective 1: Multiply Higher Roots Multiply. ) 6 2 _ 12 6) 7) _ 9 m 2 9m 2 8) _ 11 h 9) a 2 b a 2 b 10) t 2 u Mixed Exercises: Objectives 2 and Simplify completely. 9. Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. _ 20 _ 11h 2 u Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 11) _ 6 = Factor. 8 7 = 8 7 Product rule = 2 7 Simplify. 12) _ 80 = _ Factor. 16 _ Product rule = 16 = 2 Simplify. 1) _ 2 2 1) _ ) _ ) _ ) _ 2 18) _ 88 2 _ 11 19) _ ) 108 t 21) _ ) 162 _ 1 2) ) _ ) 26) _ _ 27) _ ) _ ) _ 2 _ 0) Objective : Simplify Radicals Containing Variables Simplify completely. 1) 6 d d 2 2) g 9 g ) n 20 n ) 6 t t 9 ) x y 1 xy 6) 6 a 12 b 6 a 2 b _ 7) w 1 w w 2 8) b 19 b 6 b 9) y 9 y 2 _ y 0) m 7 _ m m 1) _ d d 2 d 2) c 29 c 9 c 2 _ ) u 10 v 1 u v _ u ) x 9 y 16 x y y _ ) b 16 c _ b c bc 2 6) r 1 s 9 _ r s 2 r s 7) m n 18 n m n 2 8) a 11 b a a 2 b 9) 2x 10 y 12 2x y _ x 0) y 10 z 2 y z 8 _ 2y _ 1) 72t 17 u 7 2t u 2 9t 2 u 2) 20w x 16 wx 2wx _ m ) 8 81 _ m2 ) _ 16 x 12 2 x _ 2a ) 2 _ b 1 2a _ a h 6) b 17 12k 21 h _ h 2 k 7 _ t 7) 9 _ 81s 2 2 t t s 6 8) 2c 9 d 20 2c c d _ u 9) 28 _ v 9 u _ u m 60) v 1 _ n 8 m m n 2 72 CHAPTER 9 Radicals and Rational Exponents

47 Perform the indicated operation, and simplify. 61) ) _ ) 9 2 _ 12 6) ) _ 20 2 _ 10 66) _ ) m m m 68) t _ t t 2 _ 69) k 7 k 9 k 70) a 9 a 11 a 71) r 7 r _ r r 2 72) y 2 y 17 y 6 y _ 7) p 1 p 9 _ p p 7) c 17 c 9 c c _ 7) 9z 11 z 8 z 6 _ z 76) 2h h 16 2h 6 6 h 2 h 77) 1 _ h 2 a h 78) 20 _ a 1 a2 c 79) 11 _ c c2 z c 80) 16 _ z z z 2 _ 162d 81) 21 2d 2 d 8t d 82) 11 _ t 6 2t _ t Objective : Multiply and Divide Radicals with Different Indices The following radical expressions do not have the same indices. Perform the indicated operation, and write the answer in simplest radical form. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 8) a a = a 1/2 a / 2/ = a a / / = a = a = a a Change radicals to fractional exponents. Rewrite exponents with a common denominator. Add exponents. Rewrite in radical form. Simplify. 8) r r 2 = r / 2/ r Change radicals to fractional exponents. = r 12/1 r 10/1 Rewrite exponents with a common denominator. = r 22/1 Add exponents. = 1 _ r 22 Rewrite in radical form. 1 = r r 7 Simplify. 8) p 6 p p 86) y 2 _ 12 y y 11 87) n n n n 88) k 10 k k k 89) c c 2 1 c c 90) a 2 a 2 1 a a 91) _ w w m w 92) m m h 9) h 2 12 t h 9) _ 1 2 t 2 t 9) A block of candle wax in the shape of a cube has a volume of 6 in. The length of a side of the block, s, is given by s = V, where V is the volume of the block of candle wax. How long is each side of the block? in. 96) The radius r(v) of a sphere is a function of its volume V and can be described by the function _ V r(v) = _. If a spherical water tank has a volume π 26π of _ ft, what is the radius of the tank? ft R1) In Objective 2, you learned two methods for simplifying the nth root of a number: one uses the prime factorization of the radicand and the other involves writing the radicand as the product of two factors, one of which is a perfect nth power. Which method do you prefer and why? R2) Explain the connection between the rules for simplifying square root expressions and the rules for simplifying higher root expressions. SECTION 9. Simplifying Expressions Containing Higher Roots 7

48 9. Adding, Subtracting, and Multiplying Radicals What are your objectives for Section 9.? 1 Add and Subtract Radical Expressions 2 Simplify Before Adding and Subtracting Multiply a Binomial Containing Radical Expressions by a Monomial Multiply Radical Expressions Using FOIL Square a Binomial Containing Radical Expressions 6 Multiply Radical Expressions of the Form (a + b)(a b) How can you accomplish each objective? Write the definition of like radicals in your notes. Write the property for Adding and Subtracting Radicals in your notes. Complete the given examples on your own. Complete You Trys 1 and 2. Write the procedure for Adding and Subtracting Radicals in your own words. Complete the given example on your own. Complete You Try. Write a procedure that outlines how to multiply in this objective. Complete the given example on your own. Complete You Try. Compare using FOIL in Chapter 6 to using it for binomials containing radicals, and note what is similar and what is different. Complete the given example on your own. Complete You Try. Review the formulas for squaring a binomial in Chapter 6, if necessary. Complete the given example on your own. Complete You Try 6. Use the same formula derived in Chapter 6. Complete the given example on your own. Complete You Try 7. Read the explanations, follow the examples, take notes, and complete the You Trys. Just as we can add and subtract like terms such as x + 6x = 10xwe can add and subtract like radicals such as + 6. Definition Like radicals have the same index and the same radicand. Some examples of like radicals are and 6, and 8, x and 7 x, 2 a 2 b and a 2 b In this section, assume all variables represent nonnegative real numbers. 7 CHAPTER 9 Radicals and Rational Exponents

49 1 Add and Subtract Radical Expressions Where have you seen similar definitions and procedures before? Property Adding and Subtracting Radicals In order to add or subtract radicals, they must be like radicals. We add and subtract like radicals in the same way we add and subtract like terms add or subtract the coefficients of the radicals and multiply that result by the radical. We are using the distributive property when we combine like terms in this way. EXAMPLE 1 In-Class Example 1 Perform the operations, and simplify. a) 9x + 2x b) c) 2 2 d) Answer: a) 11x b) 11 c) 2 d) When adding or subtracting like radicals, the radical does not change. [ YOU TRY 1 ] Perform the operations, and simplify. a) x + 6x b) + 6 c) 9 d) a) First notice that x and 6x are like terms. Therefore, they can be added. x + 6x = ( + 6)x = 10x Distributive property Simplify. Or, we can say that by just adding the coefficients, x + 6x = 10x. b) Before attempting to add and 6 we must be certain that they are like radicals. Since they are like, they can be added. + 6 = ( + 6) Distributive property = 10 Simplify. Or, we can say that by just adding the coefficients of, we get + 6 = 10. c) 9 = 1 9 = (1 9) = 8 d) The radicands in are different, so these expressions cannot be combined. Perform the operations, and simplify. a) 9c + 7c b) 9 _ _ 10 c) 6 d) 6 2 EXAMPLE 2 In-Class Example 2 Perform the operations, and simplify. a) b) 9 y + 2 y y + 2 y Answer: a) b) 6 y + y Perform the operations, and simplify. 6 x + 11 x + 2 x 6 x Begin by noticing that there are two different types of radicals: x and x. Write the like radicals together. 6 x + 11 x + 2 x 6 x = 6 x + 2 x + 11 x 6 Commutative property x = (6 + 2) x + (11 6) x Distributive property = Is 8 x + x 8 x + x in simplest form? Yes. The radicals are not like (they have different indices) so they cannot be combined further. Also, each radical, x and x, is in simplest form. [ YOU TRY 2 ] Perform the operations, and simplify. 8 _ 2n _ 2n + _ 2n + _ 2n SECTION 9. Adding, Subtracting, and Multiplying Radicals 7

50 2 Simplify Before Adding and Subtracting Sometimes it looks like two radicals cannot be added or subtracted. But if the radicals can be simplified and they turn out to be like radicals, then we can add or subtract them. Procedure Adding and Subtracting Radicals 1) Write each radical expression in simplest form. 2) Combine like radicals. EXAMPLE In-Class Example Perform the operations, and simplify. a) _ 20 + _ b) 6 + _ 96 c) 0n 6 2n d) x y + y Answer: a) b) 9 6 c) _ 1 2n d) (x + 1) + y Write neatly, and write out each step very carefully to avoid making mistakes. After you complete one step, stop, look at what you have done, and ask yourself, What can I do next? Perform the operations, and simplify. a) _ 0 _ b) 7 _ 0 + c) 10 8 t _ 9 2t d) xy 6 + x 7 a) The radicals 8 2, _ 0, and are not like. The first radical is in simplest form, but 0 and _ should be simplified to determine if any of the radicals can be combined _ 0 _ = _ 9 Factor. = _ 9 Product rule = Simplify radicals. = Multiply. = 2 2 Add like radicals. b) 7 _ 0 + = is a perfect cube. = Product rule = _ 8 = 2 = 1 + Multiply. = 1 Add like radicals. c) The radical 2t is simplified, but 8t is not. We must simplify _ 8t : 8t = _ 8 _ t = _ 2 _ t = 2 2 t = 2 2t Substitute 2 2t for 8t in the original expression. 10 8t 9 2t = 10(2 2t ) 9 2t Substitute 2 2t for 8t. = 20 2t 9 2t Multiply. = 11 2t Subtract. _ d) Each radical in the expression xy 6 + x 7 must be simplified. _ xy 6 = x y 6 = x y 2 = y 2 x x 7 = x 2 x 1 7 gives a quotient of 2 and a remainder of 1. _ xy6 + x 7 2 = y 2 x + x x Substitute the simplified radicals in the original expression. = (y 2 + x 2 ) x Factor out x from exch term. In this problem we cannot add y 2 x + x 2 x like we added radicals in previous examples, but we can factor out x. (y 2 + x 2 ) x is the completely simplified form of the sum. 76 CHAPTER 9 Radicals and Rational Exponents

51 [ YOU TRY ] Perform the operations, and simplify. a) 7 _ 12 b) _ 2 6 _ _ 2 21 c) _ + _ 16 d) _ 2 6k + _ k e) mn mn In the rest of this section, we will learn how to simplify expressions that combine multiplication, addition, and subtraction of radicals. Multiply a Binomial Containing Radical Expressions by a Monomial EXAMPLE In-Class Example Multiply and simplify. a) ( 7 + _ 6 ) b) ( 6 + ) c) m ( m + 18n ) Answer: a) 12 7 b) 2 + _ 1 c) m + 2mn Use a pen with multiple colors to perform the steps in different colors as it is done in the examples. Multiply and simplify. a) _ ( _ 20 ) b) 2 _ ( 10 + _ 1 ) c) x ( x + _ 2y ) a) Because _ 20 can be simplified, we will do that first. _ 20 = _ = _ = 2 Substitute 2 for _ 20 in the original expression. ( 20 ) = ( _ 2 ) Substitute _ 2 for 20. = ( ) Subtract. = b) Neither _ 10 nor _ Multiply. 1 can be simplified. Begin by applying the distributive property. _ 2 _ ( 10 + _ 1 ) = _ 2 _ 10 + _ 2 _ 1 Distribute. = _ 20 + _ 0 Product rule Is _ 20 + _ 0 in simplest form? No. _ 20 can be simplified. = _ + _ 0 = _ + _ 0 = 2 + _ 0 c) Since 2y can be simplified, we will do that first. 2y = _ 2 y = 16 2 y = 16 _ 2 y = 2y Substitute _ 2y for 2y in the original expression. x ( x + _ 2y ) = x ( x + 2y ) Substitute 2y for _ 2y. = x x + x _ 2y Distribute. = x + 2xy Multiply. [ YOU TRY ] Multiply and simplify. a) _ 6( 7 + _ 2 ) b) _ ( + _ 21 ) c) c ( c _ 100d ) SECTION 9. Adding, Subtracting, and Multiplying Radicals 77

52 Multiply Radical Expressions Using FOIL In Chapter 6, we first multiplied binomials using FOIL (First Outer Inner Last). (2x + )(x + ) = 2x x + 2x + x + F O I L = 2x 2 + 8x + x + 12 = 2x x + 12 We can multiply binomials containing radicals the same way. EXAMPLE In-Class Example Multiply and simplify. a) ( + )(6 + ) b) ( 2 + )(2 2 2 ) c) ( + 2t )( + 2t ) Answer: a) b) 6 _ 6 10 c) + _ 10t + 6t Multiplication of binomials containing radicals uses the same procedure as multiplying binomials in Chapter 6! [ YOU TRY ] Multiply and simplify. a) (2 + _ )( + _ ) b) (2 _ + _ 2 )( _ _ 2 ) c) ( _ r + _ s )( _ r + 8 _ s ) a) Since we must multiply two binomials, we will use FOIL. (2 + )( + ) = F O = 8 + I 2 + L + Multiply. = Combine like terms. b) _ (2 + _ 2 _ )( _ 2 ) F O I L = ( 2 ) ( 2 ) = 2 + ( 10 6 ) ( 2) Multiply. = Multiply. = c) _ 9 ( r + 6 s _ )( r + 8 Combine like termes. s ) F O I L = _ r _ r + _ r 8 s + s _ r + s 8 s = r + 8 _ rs + _ rs + 8 s Multiply. = r + _ 8 rs + _ rs + 2s Multiply. = r + _ 9 rs + 2s Combine like terms. Multiply and simplify. a) (6 _ 7 )( + _ 7 ) b) ( _ 2 + _ )( _ 2 + _ ) c) ( _ 6p _ 2q )( _ 6p _ 2q ) Square a Binomial Containing Radical Expressions Recall, again, from Chapter 6, that we can use FOIL to square a binomial or we can use these special formulas: (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 78 CHAPTER 9 Radicals and Rational Exponents

53 For example, (k + 7) 2 = (k) 2 + 2(k)(7) + (7)2 and (2p ) 2 = (2p) 2 2(2p)() + () 2 = k 2 + 1k + 9 = p 2 20p + 2 To square a binomial containing radicals, we can either use FOIL or we can use the formulas above. Understanding how to use the formulas to square a binomial will make it easier to solve radical equations in Section 9.7. EXAMPLE 6 In-Class Example 6 Multiply and simplify. a) ( + 2)2 b) _ (2 t 7)2 Answer: a) 9 + b) t 28 _ t + 9 Write out every step on your paper; you will be less likely to make mistakes. [ YOU TRY 6 ] Multiply and simplify. a) ( _ 10 + ) 2 b) (2 x 6) 2 a) Use (a + b) 2 = a 2 + 2ab + b 2 _. ( 10 + ) 2 = _ ( 10 ) 2 + _ 2( 10 )() + () 2 = 10 + _ Multiply. = 19 + _ 6 10 Substitute _ 10 for a and for b. Combine like terms. b) Use (a b) 2 = a 2 2ab + b 2. (2 x 6) 2 = (2 x ) 2 2(2 x )(6) + (6) 2 = ( x) ( x )(6) + 6 Multiply. = x 2 x + 6 Multiply. Multiply and simplify. a) ( 6 + ) 2 b) ( 2 ) 2 c) ( _ w + 11 ) 2 Substitute 2 x for a and 6 for b. 6 Multiply Radical Expressions of the Form (a + b)(a b) We will review one last rule from Chapter 6 on multiplying binomials. We will use this in Section 9.6 when we divide radicals. (a + b) (a b) = a 2 b 2 For example, (t + 8)(t 8) = (t) 2 (8) 2 = t 2 6. The same rule applies when we multiply binomials containing radicals. EXAMPLE 7 In-Class Example 7 Multiply and simplify. a) ( + 7 )( 7 ) b) ( 2 )( + 2 ) c) _ ( r + )( _ r ) Answer: a) 2 b) c) r Multiply and simplify (2 x + y ) (2 x y ). Use (a + b)(a b) = a 2 b 2. (2 x + y )(2 x y ) = (2 x ) 2 ( y ) 2 Substitute 2 x for a and y for b. = (x) y Square each term. = x y Simplify. SECTION 9. Adding, Subtracting, and Multiplying Radicals 79

54 Note When we multiply expressions of the form (a + b)(a b)containing square roots, the radicals are eliminated. This will always be true. [ YOU TRY 7 ] Multiply and simplify. a) ( + _ 10 )( _ 10 ) b) ( _ h 2 _ k )( _ h + 2 _ k ) ANSWERS TO [YOU TRY] EXERCISES 1) a) 16c b) _ c) d) 6 2 2) 1 _ 2n + _ 2 2n ) a) b) _ 2 21 c) 1 2 d) _ 1 6k e) (n 2 + ) mn ) a) 2 b) + 7 c) c 2 10 _ cd ) a) b) _ 10 c) 6p 8 pq + 6q 6) a) b) 2 2 c) w w ) a) 6 b) h k 9. Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Assume all variables represent nonnegative real numbers. Objective 1: Add and Subtract Radical Expressions 1) How do you know if two radicals are like radicals? They have 2) Are the same index and 7 and the same radicand. like radicals? Why or why not? No. The indices are different. Perform the operations, and simplify. ) ) ) ) ) 6 _ 1 + _ _ 1 8) ) 1 z 2 20 z 2 z 2 10) 7 p p p 11) 2 n n 2 11 n 2 + n 2 9 n n 2 12) _ s _ s + 2 _ s + _ s 9 _ s _ s 1) _ c 8 _ 6c + _ c + 6 _ 6c _ 2 c _ 2 6c 1) _ 10 2m + _ 6 m _ 2m + _ 8 m _ 9 2m + _ 1 m Objective 2: Simplify Before Adding and Subtracting 1) What are the steps for adding or subtracting radicals? 16) Is _ 10 in simplified form? Explain. Yes. The radicals are simplified, and they are unlike. Perform the operations, and simplify. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 17) _ 8 + = _ 16 + _ Factor. = 16 _ + _ Product rule = + Simplify. = Add like radicals. 18) 8 11 = _ 11 _ 8 11 Factor. _ = Product rule = Simplify. = 6 11 Subtract like radicals. 19) 6 _ 12 20) _ ) _ ) _ 2 + _ ) _ 12 + _ 7 6 2) _ 96 + _ 2 _ 2) _ ) _ _ _ 11 27) 6 _ ) 11 _ ) 6q q + 7 q _ 1q q 0) 11 m + 8m m 1) d 2 d 2 d 2) 16k 19m m k 1 k 9 20d 2 ) 9t _ d _ t t 10 t _ t ) 8r _ r 16 _ k k r 1 8r _ r _ ) a a 7 + a 11 6a 2 _ a 6) c c 2 c 7) _ 2 8p _ 6 2p _ 2 2p 8) _ 6t + c 2 c 6 7t 18 7t 80 CHAPTER 9 Radicals and Rational Exponents

55 _ 9) 7 81a + a a 2 2a a 2 0) 0x 12 _ x 6 _ x _ 1) xy + y xy y xy 2) _ a ab + 2 a b _ 7a ab _ ) 6c 2 8d 9d 2c d c _ 2 d 2d _ ) 11v u 2u uv 2 _ uv u ) 8p 2 q 11pq 2 + p 2 88pq _ 1p 2 q 11pq 2 6) 18a 7a 2 b + 2a 7a 8 b 20a _ 7a 2 b 7) 1cd _ 9cd 9c d 1cd 9cd 8) 7yz 2 11y z + z 11y 8 z 10y 2 z 2 _ 11z 9) a 9 b b 7 0) c 8 + c 2 d b (a b 2 ) c 2 (c2 + d) Objective : Multiply a Binomial Containing Radical Expressions by a Monomial Multiply and simplify. 1) (x + ) x + 1 2) 8(k + ) 8k + 2 ) 7( 6 + 2) ) ( 7 ) 20 7 ) _ 10 ( 1) 6) 2 ( ) 0 _ 7) _ 10 6( ) 8) _ 10( _ ) ) _ 2 ( _ 20 ) 60) _ ( 18 + _ 0 ) 61) _ ( 2 _ ) _ 0 62) 2 _ ( 20 + _ 2 2 ) _ 6) ( _ 27 ) 6) ( ) 6) _ t _ ( t 81u ) 66) _ s _ 2 _ ( 12r s ) t 67) _ 9 tu 2xy _ ( 2y _ y x ) 2y x _ 2 rs + s 7 xy 2y 68) _ ab _ ( a + 27b ) _ a b + _ b a 69) c 2 ( c cd) c c + c d _ 70) mn 2 ( 2m 2 n n mn 2 ) 2m n 2 n _ m 2 Mixed Exercises: Objectives 6 71) How are the problems Multiply (x + 8)(x + ) and Multiply ( + 2 )(1 + 2 ) similar? What method can be used to multiply each of them? 72) How are the problems Multiply (y ) 2 and Multiply ( 7 2) 2 similar? What method can be used to multiply each of them? 7) What formula can be used to multiply ( + 6 )( 6 )? (a + b)(a b) = a 2 b 2 7) What happens to the radical terms whenever we multiply (a + b)(a b) where the binomials contain square roots? The radicals are eliminated. Objective : Multiply Radical Expressions Using FOIL Multiply and simplify. 7) ( p + 7)(p + 6) 76) (z 8)(z + 2) p 2 + 1p + 2z 2 6z 16 Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 77) (6 + 7 )(2 + 7 ) = Use FOIL. = Multiply. = Combine like terms. 78) ( + )(1 + ) = Use FOIL. = Multiply. = 8 + Combine like terms. 79) ( 2 + 8)( 2 ) 80) ( 6 7)( 6 + 2) ) ( + 2 x )( x ) 82) 6 ( y + 9)( y + 8) x + 1 8) x + 0 ( 2 )(2 y ) 16 + y ) ( )(2 ) 22 _ 9 1 8) ( + 2 )( ) _ ) ( + )( 6 2 ) 87) _ ( 7n + 1)( n 8) 1 88) _ _ ( 2c 2 2 _ n )( 11c ) 89) ( _ 21 _ 8 7n 2 )( + _ n 8 _ 6 ) c 90) ( 22 _ 8 2c )( c ) _ ) ( _ 6p 2 q )(8 q + _ 6p ) 92) ( r + _ s )( _ s 2 r ) 2 6pq + 0p 16q 10 _ rs + s 2r Objective : Square a Binomial Containing Radical Expressions 9) ( + 1) ) (2 + ) ) ( k 6) 2 k 12 k ) (8 + _ t ) 2 97) (1 + _ v ) 2 98) t + _ 16 t + 6 ( 2z ) 2 8v 99) _ + _ 8 v + ( 11 1 ) 2 100) 0z ( + _ 0 2z ) 2 16 _ 101) 2 ( h + 7 ) 2 102) ( m _ 2 9 ) 2 h + _ 10) 2 7h + ( x 7 y ) 2 10) m ( b + _ 2 m + a ) 2 x a _ 2 ab + b 2 xy + y Objective 6: Multiply Radical Expressions of the Form (a + b)(a b) 10) (c + 9)(c 9) c ) (g 7)(g + 7) 107) (6 )(6 + ) 1 108) ( g 2 7 )( ) 9 SECTION 9. Adding, Subtracting, and Multiplying Radicals 81

56 109) ( p + 7)( p 7) 110) ( w + 10)( w 10) p 9 111) ( + 2 )( w ) 6 112) ( )( ) 20 11) ( _ 2 )( _ 2 + ) 9 11) (1 + 6 )(1 6 ) 1 _ 6 11) ( c + d )( c d ) c d 116) _ ( 2y + _ z _ )( 2y _ z ) 2y z 117) _ (8 f g _ )(8 f + g ) 6f g 118) ( a + _ b) ( a _ b) a 6b Extension Multiply and simplify. 119) (1 + 2 )(1 2 + _ 2 ) 1 120) ( + 2 )(9 2 + ) 29 Let f (x) = x 2. Find each function value. 121) f ( 7 + 2) ) f ( 6 ) ) f (1 2 ) 1 12) f ( 2 + ) R1) Explain the difference between mastering the concepts from Chapter 6 to completing these exercises. How did it compare? R2) Explain how you could substitute x for the radical and y for the radical 2 in Exercise 111 to help you multiply. R) Which exercises did you understand well, and which were more difficult for you? Identify where you had difficulty doing a problem. 9.6 Dividing Radicals What are your objectives for Section 9.6? 1 Rationalize a Denominator: One Square Root 2 Rationalize a Denominator: One Higher Root Rationalize a Denominator Containing Two Terms How can you accomplish each objective? Understand what rationalizing a denominator means. Complete the given examples on your own, and develop a procedure for rationalizing a denominator that contains one square root. Complete You Trys 1. Follow the explanation to understand the logic behind rationalizing a higher root. Complete the given examples on your own, and develop a procedure for rationalizing a denominator that contains one higher root. Complete You Trys and. Understand the definition of a conjugate, and write a few examples in your notes. Recall the formula for (a + b)(a b), and know how it can help you multiply conjugates. Learn the procedure for Rationalizing a Denominator That Contains Two Terms. Complete the given examples on your own. Complete You Trys 6 and 7. (continued) 82 CHAPTER 9 Radicals and Rational Exponents

57 What are your objectives for Section 9.6? Rationalize a Numerator Divide Out Common Factors from the Numerator and Denominator How can you accomplish each objective? Follow the same procedures developed for rationalizing the denominator, but focus instead on the numerator. Complete the given example on your own. Complete You Try 8. Know how to apply the properties of real numbers, especially factoring and distributing, to simplify. Complete the given example on your own. Complete You Try 9. Read the explanations, follow the examples, take notes, and complete the You Trys. It is generally agreed that a radical expression is not in simplest form if its denominator contains a radical. For example, 1 is not simplified, but an equivalent form,, is simplified. Later we will show that 1 =. The process of eliminating radicals from the denominator of an expression is called rationalizing the denominator. We will look at two types of rationalizing problems. 1) Rationalizing a denominator containing one term 2) Rationalizing a denominator containing two terms To rationalize a denominator, we will use the fact that multiplying the numerator and denominator of a fraction by the same quantity results in an equivalent fraction: 2 = _ and _ 8 12 are equivalent because = 1 We use the same idea to rationalize the denominator of a radical expression. 1 Rationalize a Denominator: One Square Root The goal of rationalizing a denominator is to eliminate the radical from the denominator. With regard to square roots, recall that a a = a 2 = afor a 0. For example, _ = (19)2 = 19 and _ t _ t = t 2 = t (t 0) We will use this property to rationalize the denominators of the following expressions. EXAMPLE 1 In-Class Example 1 Rationalize the denominator of each expression. 2 a) _ b) _ 20 c) Answer: a) 2 _ b) 2 c) Rationalize the denominator of each expression. 1 a) 6 b) _ 18 c) _ 2 SECTION 9.6 Dividing Radicals 8

58 a) To eliminate the square root from the denominator of 1, ask yourself, By what do I multiply to get a perfect square under the square root? The answer is because = 2 = 9 =. Multiply by in the numerator and denominator. (We are actually multiplying by 1.) 1 = 1 = 2 = = 9 Rationalize the denominator. BE CAREFUL is in simplest form. We cannot simplify terms inside and outside of the radical. = 1 = 1 = 1 Incorrect! 1 Be sure you understand why you multiply by a particular radical to rationalize the denominator! [ YOU TRY 1 ] b) First, simplify the denominator of = 6 2 = _ _ = _ _ 2 = 12 2 = Simplify Simplify. Rationalize 18. the denominator. c) To rationalize _, multiply the numerator and denominator by 2. 2 _ = _ 2 _ 2 2 _ 2 = _ 6 2 Rationalize the denominator of each expression. 1 a) 1 b) _ 7 27 c) _ 9 6 Sometimes we will apply the quotient or product rule before rationalizing. EXAMPLE 2 In-Class Example 2 Simplify completely. _ 6 a) _ 20 b) _ _ 9 Answer: a) 0 10 b) 6 Simplify completely. a) _ 2 b) _ 1 7 a) Begin by simplifying the fraction _ under the radical. 2 2 = 1 8 Simplify. 1 = = 1 8 = 1 2 = _ _ _ 2 = = 2 8 CHAPTER 9 Radicals and Rational Exponents

59 [ YOU TRY 2 ] b) Begin by using the product rule to multiply the radicands. _ 1 7 = _ 1 7 Product rule Multiply the fractions under the radical. = 1 _ 7 1 = 2 6 Multiply. = = _ _ 6 = 0 6 Simplify completely. 10 a) _ b) _ We work with radical expressions containing variables the same way. In the rest of this section, we will assume that all variables represent positive real numbers. EXAMPLE In-Class Example Simplify completely. _ 7 a) m b) 27p q _ 10xy c) x y 2 Answer: a) 7 _ m m b) p 2 _ q c) _ 10xy q 2 x 2 y Ask yourself the questions found in parts a) and b). This will help you understand the next objective too. Simplify completely. 2 a) _ x b) _ 12m 7n c) 6cd 2 cd a) Ask yourself, By what do I multiply x to get a perfect square under the square root? The perfect square we want to get is x 2. x? = x 2 = x x _ x = x 2 = x 2 x = 2 x x x = 2 _ x = _ 2 x x 2 x Rationalize the denominator. b) Before rationalizing, apply the quotient rule and simplify the numerator. _ 12m 7n = 12 m = 2m m 7n 7n Rationalize the denominator. By what do I multiply _ 7n to get a perfect square under the square root? The perfect square we want to get is 7 2 n 2 or 9 n 2. _ 7n? = 7 2 n 2 _ = 7n 7n 7n = 7 2 n 2 = 7n _ 12m 7n = 2m m 7n _ 2m m = 7n 7n 7n = 2m _ 21mn 7n Rationalize the denominator. SECTION 9.6 Dividing Radicals 8

60 [ YOU TRY ] c) _ 6cd 2 cd = 6 d Simplify the radicand using the quotient rule for exponents. 6 = d = 6 d d d = 6d d Simplify completely. a) p b) 18k 10m _ c) _ 20r s s 2 2 Rationalize a Denominator: One Higher Root Many students assume that to rationalize all denominators we simply multiply the numerator and denominator of the expression by the denominator as in = = _. We will see, however, why this reasoning is incorrect. To rationalize an expression like we asked ourselves, By what do I multiply to get a perfect square under the square root? To rationalize an expression like _ we must ask ourselves, By what do I multiply 2 2 to get a perfect cube under the cube root? The perfect cube we want is 2 (since we began with 2) so that = 2 = 2. We will practice some fill-in-the-blank problems to eliminate radicals before we move on to rationalizing. EXAMPLE In-Class Example Fill in the blank. a) 6 _? = _ 6 _ = 6 b) 9? = _ 7 = 7 c)? m = _ m = m d)? = _ 2 = 2 e)? = 2 = 2 Answer: a) 6 2 b) 7 c) m 2 d) 2 2 e) 2 Get in the habit of asking yourself these questions! Fill in the blank. a) _? = = b) c) x 2? = x = x d) 8 e) 27? = = a) Ask yourself, By what do I multiply to get? = =? = 2 = 2? The answer is? = = 2 = = b) By what do I multiply to get? 2? = = 2 = = c) By what do I multiply x 2 to get x? x x 2? = x = x x 2 x = x = x _ CHAPTER 9 Radicals and Rational Exponents

61 Don t move to the next example unless you have fully grasped this concept first! [ YOU TRY ] d) In this example, 8? = 2 = 2,why are we trying to obtain 2 instead of 8?Because in the first radical, 8, 8 is a power of 2. Before attempting to fill in the blank, rewrite 8 as 2. 8? = 2 = 2 2? = 2 = = 2 = 2 e) _ 27? = =? = = Since 27 is a power of, rewrite 27 as. = = Fill in the blank. a) 2? = 2 = 2 b) c) 12? = = t 2? = t = t We will use the technique presented in Example to rationalize denominators with indices higher than 2. EXAMPLE In-Class Example Rationalize the denominator. 8 a) b) 2 9 c) _ 9 _ t 8 _ 9 Answer: a) _ b) 18 9 c) _ t t Be sure to ask yourself this question so that you multiply by the correct radical! Rationalize the denominator. 7 a) _ b) c) _ 7 n a) First identify what we want the denominator to be after multiplying. We want to obtain since =. _ 7 = This is what we want to get. What is needed here? Ask yourself, By what do I multiply to get? 2 _ = 7 _ 2 Multiply. 7 9 = Simplify. b) Use the quotient rule for radicals to rewrite as _. Then, write as 2 2 to get _ = 2 2 What denominator do we want to get after multiplying? We want to obtain 2 since 2 = = 2 This is what we want to get. What is needed here? SECTION 9.6 Dividing Radicals 87

62 In your own words, summarize how to rationalize denominators in Objective 2. [ YOU TRY ] By what do I multiply 2 2 to get 2? 2 2 = = _ = 2 _ 2 Multiply. 2 Multiply. c) What denominator do we want to get after multiplying? We want to obtain n since n = n. _ 7 n = n This is what we want to get. What is needed here? Ask yourself, By what do I multiply n to get n? n Rationalize the denominator. a) _ b) _ c) _ 8 w _ 7 n n = 7 _ n n n Multiply. = 7 _ n n Simplify. Rationalize a Denominator Containing Two Terms 1 To rationalize the denominator of an expression like _ +, we multiply the numerator and the denominator of the expression by the conjugate of +. Definition The conjugate of a binomial is the binomial obtained by changing the sign between the two terms. Expression Conjugate a + b a b In Section 9. we applied the formula (a + b)(a b) = a 2 b 2 to multiply binomials containing square roots. Recall that the terms containing the square roots were eliminated. 88 CHAPTER 9 Radicals and Rational Exponents

63 EXAMPLE 6 In-Class Example 6 Multiply + by its conjugate. Answer: 1 Multiply 8 6 by its conjugate. The conjugate of 8 6 is We will first multiply using FOIL to show why the radical drops out, then we will multiply using the formula (a + b)(a b) = a 2 b 2 i) Use FOIL to multiply. (8 6 )(8 + 6 ) = ii) Use (a + b)(a b) = a 2 b F O I L = 6 6 = 8 6 (8 _ 6 )(8 + _ 6 ) = (8) 2 ( _ 6 ) 2 Substitute 8 for a and _ 6 for b. = 6 6 = 8 Each method gives the same result. [ YOU TRY 6 ] Multiply 2 + _ 11 by its conjugate. Procedure Rationalize a Denominator Containing Two Terms If the denominator of an expression contains two terms, including one or two square roots, then to rationalize the denominator we multiply the numerator and denominator of the expression by the conjugate of the denominator. EXAMPLE 7 In-Class Example 7 Rationalize the denominator, and simplify completely. a) _ 6 + b) _ c d 2 d + c Answer: a) 18 _ 2 _ cd c c + d b) d + cd d c 2 Rationalize the denominator, and simplify completely. a) _ + b) _ a + b b a a) The denominator of _ + has two terms, so we multiply the numerator and denominator by, the conjugate of the denominator. + Multiply by the conjugate. = ( _ ) () 2 ( ) 2 (a + b)(a b) = a2 b 2 = 1 2 Simplify. = 1 Subtract SECTION 9.6 Dividing Radicals 89

64 [ YOU TRY 7 ] b) a + b b a b + a Multiply by the conjugate. b + a In the numerator we must multiply ( a + b)( b + a).we will use FOIL. a + b b a _ b + a b + a = ab + a a + _ b b + ab FOIL in the numerator. Use ( b ) 2 (a) 2 (a + b)(a b) = a 2 b 2 _ ab + a a + b = _ b + ab b a 2 Square the terms. Rationalize the denominator, and simplify completely. 1 a) _ 7 2 b) c _ d c + d Rationalize a Numerator In higher-level math courses, sometimes it is necessary to rationalize the numerator of a radical expression so that the numerator does not contain a radical. EXAMPLE 8 In-Class Example 8 Rationalize the numerator, and simplify _ completely. a) _ 11 b) _ Answer: a) _ b) _ Follow the same process you did for rationalizing the denominator, but focus on the numerator. [ YOU TRY 8 ] Rationalize the numerator, and simplify completely. 7 a) 8 b) _ 2 a) Rationalizing the numerator of 7 means eliminating the square root from the 2 numerator. Multiply the numerator and denominator by.7 7 = 7 2 _ 7 2 _ 7 = 7 _ 1 b) To rationalize the numerator we must multiply the numerator and denominator by 8 +,the conjugate of the numerator Multiply by the conjugate. = 82 _ ( ) 2 (8 + ) (a + b)(a b) = a2 b2 Multiply. = = Rationalize the numerator, and simplify completely. a) 6 + b) _ 7 90 CHAPTER 9 Radicals and Rational Exponents

65 Divide Out Common Factors from the Numerator and Denominator Sometimes it is necessary to simplify a radical expression by dividing out common factors from the numerator and denominator. This is a skill we will need in Chapter 10 to solve quadratic equations, so we will look at an example here. EXAMPLE 9 Simplify completely: _ In-Class Example Simplify completely: _. 9 Answer: 2 BE CAREFUL It is tempting to do one of the following: + 12 = + 12 Incorrect! or + 12 = + Incorrect! Each is incorrect because is a term in a sum and 12 is a term in a sum. We will use this skill in Chapter 10, so be sure you understand it. The correct way to simplify _ + 12 is to begin by factoring out a in the numerator and then divide the numerator and denominator by any common factors ( = _ + ) Factor out from the numerator. 1 ( + ) = Divide by. 1 = + Simplify. We can divide the numerator and denominator by in ( _ + ) because the in the numerator is part of a product, not a sum or difference. [ YOU TRY 9 ] Simplify completely a) _ b) 20 _ 6 2 ANSWERS TO [YOU TRY] EXERCISES 1) a) 7 7 b) _ c) 9 _ 0 2) a) 1 7 b) 1 p ) a) _ p b) k2 _ km m c) 2r rs ) a) 2 2 or b) t c) ) a) _ 9 6 b) _ s 7 c) 8w 2 6) 7 w 7) a) _ b) c2 _ 2c d + d 8) a) _ c 2 d 1 b) _ ) a) b) 10 _ SECTION 9.6 Dividing Radicals 91

66 9.6 Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Assume all variables represent positive real numbers. Objective 1: Rationalize a Denominator: One Square Root 1) What does it mean to rationalize the denominator of a radical expression? Eliminate the radical from the denominator. 2) In your own words, explain how to rationalize the denominator of an expression containing one term in the denominator. Answers may vary. Rationalize the denominator of each expression. ) 1 ) ) 8 9) _ 28 _ 20 11) _ 60 1) 6 8 ) 1 6 _ 6 6) _ _ ) _ 6 _ ) _ 8 27 _ _ 12) _ ) 66 _ 12 _ 22 2 Multiply and simplify. _ 10 1) _ 7 7 _ 0 16) 11 _ ) 1 _ 8 _ ) 11 _ 10 _ 8 11 Simplify completely. 8 19) _ y 21) _ t 8 y _ y 20) t t _ w 22) 2 m _ w w 2m m 22 _ 2 2 _ 6v 2) 7 w 8v _ vw 2) w 81c 2d _ 9c2 2cd 2d 2) _ a b ab a _ b b 26) m _ 2 n _ n2 7m 7m n 7m _ 27) 7 _ _ b 28) 2 b b 2 _ 2 6v v v 2 29) 1 1j j j 0) 22 w 7 22w w Objective 2: Rationalize a Denominator: One Higher Root Fill in the blank. 1) 2? = 2 = 2 2)? = = 2 2 ) or 9? = 2 = ) or 2? = 2 = 2 ) c? = c 2 = c 6) p? = p = p c 2 7) p 2? = 2 = 2 2 or 8 8) _ 16? = 2 = 2 2 _ 9) m _? = m = m m 0) k? = k = k k 1) Inez simplifies _ 8 8 like this: _ _ = 8. Is she right or wrong? If her answer is wrong, explain her mistake and do the problem correctly. Her answer is wrong because = 9. The right way to simplify _ 8 8 is _ _ 9 8 = _ = ) Stavros simplifies 8 like this: 8 x 8 x x 8 x = 8 _ x. x Is he right or wrong? If his answer is wrong, explain his mistake and do the problem correctly. _ 8 His answer is right: x 8 x 8 = x 8 = x. Rationalize the denominator of each expression. ) _ ) 12 _ _ 2 7) 9) 9 1) 8 ) _ 10 _ z ) n 2 9 ) _ _ 2 6) _ ) _ ) _ 2 _ 0 _ ) _ z 2 ) _ 6 z u 6 _ u 2 u _ n n 6) x 2 _ x x 7 7) _ 28k 2k 2 _ 2k 8) _ 2 _ 2t 2 10t t 9 9) 9 a 2 a _ a 60) 8 8 h h 2 _ h _ 61) _ 2m 0m 2m 62) 2 _ t 2 2 t t 92 CHAPTER 9 Radicals and Rational Exponents

67 Objective : Rationalize a Denominator Containing Two Terms 6) How do you find the conjugate of an expression with two radical terms? Change the sign between the two terms. 6) When you multiply a binomial containing a square root by its conjugate, what happens to the radical? The radical is eliminated. Find the conjugate of each expression. Then, multiply the expression by its conjugate. 6) ( + 2 ) ( 2 ); 2 66) ( ) ( + ); 11 67) ( ) ( _ 10 ) ( 2 69) _ 6 ); ( ( t 8) 70) + _ 10 ); 7 _ ( p + ) ( t + 8); t 6 ( p ); p 2 Rationalize the denominator, and simplify completely. = ) Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 6 71) _ = _ 6 + _ + Multiply by the 6( + conjugate. ) = _ () 2 ( ) 2 (a + b) (a b) = a 2 b = Simplify ) _ 7 + = _ _ ( 7 2 ) = _ ( 7 ) 2 ( 2 ) 2 = _ = _ 2 _ 2 7) _ ) 10 7) _ ) ) _ ) 2 Multiply terms in numerator; square terms in denominator. Multiply by the conjugate. (a + b) (a b) = a 2 b 2 Multiply terms in numerator; square terms in denominator. Simplify. _ _ 1 _ + _ _ 2 10 _ ) 10 80) _ m 81) _ m + n m mn 82) _ u m n u v u + _ uv u v 8) _ b 2 b d 9 b + 8) _ d + d 8) x + y x y f g 86) _ f + g f _ 2 fg + g f g Objective : Rationalize a Numerator Rationalize the numerator of each expression, and simplify. 87) _ 88) 2 9 _ ) x x 7 7x 90) 8a a b 2ab 91) ) 1 + _ 7 _ ) _ x 2 x _ 1 x + 2 9) _ n n n 9) c ) x + h x c h 97) Does rationalizing the denominator of an expression change the value of the original expression? Explain your answer. 98) Does rationalizing the numerator of an expression change the value of the original expression? Explain your answer. Objective : Divide Out Common Factors from the Numerator and Denominator Simplify completely. 99) + 10 _ ) 18 6 _ ) _ 102) ) _ + 6 _ ) ) 10 _ ) + _ SECTION 9.6 Dividing Radicals 9

68 A 107) The function r(a) = describes the radius of a circle, r(a), in terms of its area, π A. a) If the area of a circle is measured in square inches, find r(8π) and explain what it means in the context of the problem. b) If the area of a circle is measured in square inches, find r(7) and rationalize the denominator. Explain the meaning of r(7) in the context of the problem. c) Obtain an equivalent form of the function by rationalizing the denominator. r(a) = Aπ π _ 108) The function r(v) = _ V describes the radius of a π sphere, r(v), in terms of its volume, V. a) If the volume of a sphere is measured in cubic centimeters, find r(6π) and explain what it means in the context of the problem. b) If the volume of a sphere is measured in cubic centimeters, find r(11) and rationalize the denominator. Explain the meaning of r(11) in the context of the problem. c) Obtain an equivalent form of the function by _ rationalizing the denominator. r(v) = _ 6π 2 V 2π R1) Why is it important to always read the directions carefully? R2) Look at Exercises and. Explain how rationalizing their denominators is similar as well as how that process is different. R) Did you circle any problems you struggled with? Ask about them in class! Putting It All Together What are your objectives? 1 Review the concepts of Sections How can you accomplish each objective? Complete the given examples on your own. Complete You Try 1. Read the explanations, follow the examples, take notes, and complete the You Trys. 1 Review the Concepts Presented in In Section 9.1, we learned how to find roots of numbers. For example, 6 = because ( ) = 6. We also learned about square root and cube root functions. In Section 9.2, we learned about the relationship between rational exponents and radicals. Recall that if m and n are positive integers and m is in lowest terms, then n a m/n = (a 1/n ) m n = ( a ) m provided that a 1/n is a real number. For the rest of this section we will assume that all variables represent positive real numbers. 9 CHAPTER 9 Radicals and Rational Exponents

69 EXAMPLE 1 In-Class Example 1 Simplify completely. The answer should contain only positive exponents. a) (12)2/ b) _ c 9 d 11/6 ( c 1 d 2 ) /2 d/12 Answer: a) 2 b) _ ( 2c 2 ) Simplify completely. The answer should contain only positive exponents. a) (2) / b) a 7 b 9/8 /2 _ ( 2a 9 b / ) a) The denominator of the fractional exponent is the index of the radical, and the numerator is the power to which we raise the radical expression. b) 2 / = ( _ = (2) /2 = 16 2 ) Write in radical form. _ 2 = 2 a 7 b 9/8 /2 _ ( 2a 9 b / = _ 2a 9 b / ) ( a 7 b 9/8 ) Simplify the expression inside the parentheses by subtracting the exponents. Eliminate the negative from the outermost exponent by taking the reciprocal of the base. = (2a 9 7 b / 9/8 ) /2 = (2a 2 b 6/8 9/8 ) /2 = (2a 2 b /8 ) /2 Apply the power rule, and simplify. = (2) /2 (a 2 ) /2 (b /8 ) /2 = ( 2 ) a b 9/16 = a b 9/16 12a = _ b 9/16 In Sections we learned how to simplify, multiply, divide, add, and subtract radicals. Let s look at these operations together so that we will learn to recognize the techniques needed to perform these operations. EXAMPLE 2 In-Class Example 2 Perform the operations, and simplify. a) b) 6 ( + 2 )6 Answer: a) 7 10 b) Look carefully at expressions like these so that you can distinguish between a product and a sum or difference. Perform the operations, and simplify. a) b) (10 6 ) a) This is the sum and difference of radicals. Remember that we can only add and subtract radicals that are like radicals = _ _ Write like radicals together. = Subtract. b) This is the product of radical expressions. We must multiply the binomial 10 6 by using the distributive property. (10 6 ) = _ 10 6 _ Distribute. = _ Product rule; _ _ =. = _ Multiply. Ask yourself, Is _ in simplest form? No. _ 18 can be simplified. = _ is a perfect square. = _ Product rule = _ 9 = = Multiply. The expression is now in simplest form. Putting It All Together 9

70 Next we will look at multiplication problems involving binomials. Remember that the rules we used to multiply binomials like (x + )(x 9)are the same rules we use to multiply binomials containing radicals. EXAMPLE In-Class Example Multiply and simplify. a) (1 + )(8 7 ) b) ( k + _ 11 )( k _ 11 ) c) ( )2 Answer: a) _ 21 b) k 11 c) 2 0 Notice the connection between what you learned in Chapter 6 and multiplying binomials containing radicals. EXAMPLE In-Class Example Rationalize the denominator of each expression. 1 a) _ 2a b) _ 1 2a 1 c) _ 2 + _ Answer: a) 7 2a a 7 a 2 _ b) 1 2 c) 7 a 7 Multiply and simplify. a) (8 + 2 )(9 _ 11 ) b) ( n + 7 )( n 7 ) c) 2 (2 ) a) Since we must multiply two binomials, we will use FOIL. F O I L (8 + 2 )(9 _ 11 ) = _ _ 11 Use FOIL. = 72 8 _ _ 22 Multiply. All radicals are simplified and none of them are like radicals, so this expression is in simplest form. b) We can multiply ( n + 7 )( n 7 )using FOIL or, if we notice that this product is in the form (a + b)(a b)we can apply the rule (a + b)(a b) = a 2 b 2. Either method will give us the correct answer. We will use the second method. (a + b)(a b) = a 2 b 2 _ ( n + _ 7 _ )( n _ 7 ) = _ ( n ) 2 _ ( 7 ) 2 = n 7 Substitute _ n for a and _ 7 for b. c) Once again we can either use FOIL to expand (2 ) 2, or we can use the special formula we learned for squaring a binomial. We will use (a b) 2 = a 2 2ab + b 2. _ (2 ) 2 = _ (2 ) 2 _ 2(2 )() + () 2 Substitute 2 _ for a and for b. = ( ) () + 9 Multiply. = Multiply. = Combine like terms. Remember that an expression is not considered to be in simplest form if it contains a radical in its denominator. To rationalize the denominator of a radical expression, we must keep in mind the index on the radical and the number of terms in the denominator. Rationalize the denominator of each expression. 10 a) 2x b) 10 2x c) _ a) First, notice that the denominator of 10 contains only one term and it is a square root. 2x Ask yourself, By what do I multiply _ 2x to get a perfect square under the radical? The answer is 2x since _ 2x _ 2x = x 2 = 2x.Multiply the numerator and denominator by 2x,and simplify. 10 _ 2x = 10 _ 2x 2x 2x Rationalize the denominator. _ 10 2x _ = _ = 10 2x = 2x x 2 2x x 96 CHAPTER 9 Radicals and Rational Exponents

71 In your notes, summarize the similarities and differences in rationalizing the denominators in each expression in this example. [ YOU TRY 1 ] Read all directions first. b) The denominator of 10 _ contains only one term, but it is a cube root. Ask yourself, 2x By what do I multiply 2x to get a radicand that is a perfect cube? The answer is x 2 since _ 2x x 2 = _ 8x = 2x.Multiply the numerator and denominator by x 2, and simplify. 10 _ 2x = 10 _ 2x _ x 2 x 2 Rationalize the denominator. _ 10 x 2 = = 10 x 2 = _ x 2 8x 2x x c) The denominator of 10 contains two terms, so how do we rationalize the 2 1 denominator of this expression? We multiply the numerator and denominator by the conjugate of the denominator = Multiply by the conjugate _ 10 ( = 2 + 1) ( 2 ) 2 (1) 2 Multiply. (a + b)(a b) = a 2 b 2 _ 20 + _ 10 = Distribute. 1 Simplify. = 2 + _ = _ 2 ; simplify. a) Perform the operations, and simplify. i) 2 ( w + 8) ii) iv) _ ( _ a )( + _ 10 2 v) (2 a ) iii) 2 (9 _ 10 2 ) + y)(2 y) b) Find each root. 121 i) 16 ii) 1000 iii) 0.09 iv) 9 c) Simplify completely. The answer should contain only positive exponents. 2/ i) ( 6) ii) 81x y 1/2 / _ ( x y 6 ) d) Rationalize the denominator of each expression. i) 2 _ 9h ii) 7 + _ 6 + iii) ANSWERS TO [YOU TRY] EXERCISES 1) a) i) w + 16 w + 6 ii) 12 _ a a iii) 18 2 iv) _ 10 v) 12 y 2 11 b) i) _ ii) 10 iii) 0. iv) not a real number c) i) 16 ii) y/8 27x 6 d) i) 8 _ h 2 ii) 22 _ 6 iii) 8 7 h 10 Putting It All Together 97

72 Putting It All Together Exercises *Additional answers can be found in the Answers to Exercises appendix. Objective 1: Review the Concepts Presented in Fill in the blank with always, sometimes, or never to make the statement true. 1) The odd root of a real number is a real number. 2) The even root of a negative number is a real number. Assume all variables represent positive real numbers. Find each root, if possible. ) _ 81 ) ) _ 6 2 6) ) _ not a real number 8) 9 _ 12 7 Simplify completely. The answer should contain only positive exponents. 9) (1) 1/2 / 12 10) ( 2) 11) / ) ( _ 81 / ) Simplify completely. 19) _ ) _ ) _ ) always 16 never not a real number 1) 12 1/ 1 1) 100 ( 9 /2 ) _ ) k / k / ) (t /8 ) 16 6 t k /10 27a 8 2/ 17) _ ( b 9 ) 9 18x 9 y / /2 18) x 0 a 16/ b 6 ( 2x y ) 2y /6 Do the exercises, and check your work. 2) 2 _ 2) c 11 c _ c _ 2m 2 n 12m 26) 6x 19 y 20 2x 2x y 2) 96m 7 n 1 Perform the operations, and simplify. 27) _ 12 _ ) _ 96k 11 2k 2k2 29) (6 + 7 )(2 + 7 ) ) c 2 _ 108c 1 2c 7 18c 2 _ 1) 2 (2 + 10) ) ( 2 + )( 2 ) ) 18 6 ) _ ) 7m n + _ m 12mn 17m mn 6) 6p 7 q 1pq 2 p q 2 10q 7) 60t 8 u 2t u 8) _ 9 t 2 9 u 2 2 9) (8 _ w ) 2 w _ 16 w + 6 0) (2 x y )( x 6 y ) 1) 2 _ + a _ ) b 2 c 10x + 6y 17 _ xy 2 2a 2) r 2 _ r r 6 r 16 a _ 9c b 2 2 c 2 ) _ 2 _ 2w c w 11 w For each function, find the domain and graph the function. ) f (x) = x 2 6) g(x) = x 7) h(x) = x + 1 8) k(x) = x + 1 9) g(x) = _ x 0) f (x) = x 2 7 R1) Which types of problems did you struggle with the most? What made them difficult? R2) Were you able to easily distinguish between the methods to use for the different problems in this homework? How often did you have to look back at your notes or the book to do the problems? If you had to take a quiz or test on this material now, would you be ready? 98 CHAPTER 9 Radicals and Rational Exponents

73 9.7 Solving Radical Equations What are your objectives for Section 9.7? 1 Understand the Steps for Solving a Radical Equation 2 Solve an Equation Containing One Square Root Solve an Equation Containing Two Square Roots Solve an Equation Containing a Cube Root How can you accomplish each objective? Write the procedure for Solving Radical Equations Containing Square Roots in your own words, and be aware of extraneous solutions. Use the steps for Solving Radical Equations Containing Square Roots. Be aware that you will, often, have to square a binomial. Complete the given examples on your own. Complete You Trys 1 and 2. Use the steps for Solving Radical Equations Containing Square Roots. Recognize when you need to square both sides of the equation twice. Complete the given examples on your own. Complete You Trys. Follow the explanation to create your own procedure for solving radical equations containing a cube root. Complete the given example on your own. Complete You Try 6. Read the explanations, follow the examples, take notes, and complete the You Trys. In this section, we will learn how to solve radical equations. An equation containing a variable in the radicand is a radical equation. Some examples of radical equations are p = 7, n = 2, _ 2x = x, w + 6 w + 1 = 1 1 Understand the Steps for Solving a Radical Equation Let s review what happens when we square a square root expression: If x 0,then ( x ) 2 = x.that is, to eliminate the radical from x, we square the expression. Therefore to solve equations like those above containing square roots, we square both sides of the equation to obtain new equations. The solutions of the new equations contain all of the solutions of the original equation and may also contain extraneous solutions. An extraneous solution is a value that satisfies one of the new equations but does not satisfy the original equation. Extraneous solutions occur frequently when solving radical equations, so we must check all possible solutions in the original equation and discard any that are extraneous. SECTION 9.7 Solving Radical Equations 99

74 Procedure Solving Radical Equations Containing Square Roots Step 1: Get a radical on a side by itself. Step 2: Square both sides of the equation to eliminate a radical. Step : Combine like terms on each side of the equation. Step : If the equation still contains a radical, repeat Steps 1. Step : Solve the equation. Step 6: Check the proposed solutions in the original equation, and discard extraneous solutions. 2 Solve an Equation Containing One Square Root EXAMPLE 1 In-Class Example 1 Solve. _ a) b 8 = 2 b) w = Answer: a) {12} b) Write out the steps as you are reading them. Be sure to write neatly and in a very orderly way. Solve. a) c 2 = b) t = 0 a) Step 1: The radical is on a side by itself: c 2 = Step 2: Square both sides to eliminate the square root. ( c 2 ) 2 = 2 Square both sides. c 2 = 9 Steps and do not apply because there are no like terms to combine and no radicals remain. Step : Solve the equation. c = 11 Add 2 to each side. Step 6: Check c = 11in the original equation. c 2 _ = 11 2 =? 9 = The solution set is {11}. b) The first step is to get the radical on a side by itself. t = 0 t + = 6 Subtract 6 from each side. ( t + ) 2 = ( 6) 2 Square both sides to eliminate the radical. t + = 6 The square root has been eliminated. t = 1 Solve the equation. Check t = 1 in the original equation. _ t = 0 _ =? 0? = 0 FALSE Because t = 1 gives us a false statement, it is an extraneous solution. The equation has no real solution. The solution set is. 600 CHAPTER 9 Radicals and Rational Exponents

75 [ YOU TRY 1 ] Solve. a) a + = 7 b) _ m = 9 Sometimes, we have to square a binomial in order to solve a radical equation. Don t forget that when we square a binomial we can either use FOIL or one of the following formulas: (a + b) 2 = a 2 + 2ab + b 2 or (a b) 2 = a 2 2ab + b 2. EXAMPLE 2 In-Class Example 2 Solve _ 2y + 2 = y. Answer: { 2, 0} Review Section 6. if you need help squaring a binomial. Solve _ 2x = x. Start by getting the radical on a side by itself. _ 2x + 1 = x 1 Subtract 1 from each side. ( 2x + 1 ) 2 = (x 1) 2 Square both sides to eliminate the radical. 2x + 1 = x 2 2x + 1 Simplify; square the binomial. 0 = x2 x Subtract 2x; subtract 1. 0 = x(x ) Factor. [ YOU TRY 2 ] Solve. x = 0 or x = 0 Set each factor equal to zero. x = 0 or x = Solve. Check x = 0 and x = in the original equation. x = 0: _ 2x = x x = : _ 2x = x _ 2(0) =? 0 _ 2() =? =? =? 2 = 1 FALSE x = is a solution but x = 0is not because x = 0does not satisfy the original equation. The solution set is {}. a) p + 10 = p b) _ h h = 2? + 1 = TRUE Solve an Equation Containing Two Square Roots Next, we will take our first look at solving an equation containing two square roots. EXAMPLE In-Class Example Solve _ b + 2 b 1 = 0. Answer: {8} Solve _ 2a + a = 0. Begin by getting a radical on a side by itself. _ 2a + = a Add _ a to each side. ( 2a + ) 2 = ( a ) 2 Square both sides to eliminate the radicals. 2a + = 9(a ) 2 = 9 2a + = 9a Distribute. 7a = 9 a = 7 Solve. SECTION 9.7 Solving Radical Equations 601

76 Check a = 7in the original equation. The solution set is {7}. [ YOU TRY ] Solve r _ 6r + 2 = 0. _ 2a + _ a = 0 2(7) + 7 = 0 _ =? 0 _ 18 2 =? = 0 Recall from Section 9. that we can square binomials containing radical expressions just like we squared (x 1) 2 in Example 2. We can use FOIL or the formulas (a + b) 2 = a 2 + 2ab + b 2 or (a b) 2 = a 2 2ab + b 2 EXAMPLE In-Class Example Square and simplify (6 + r + 1 )2. Answer: r r Make sure you understand this example before going on to the next. [ YOU TRY ] Square and simplify ( _ m + 2 ) 2. Use the formula (a b) 2 = a 2 2ab + b 2. ( _ m + 2 ) 2 = () 2 2()( _ m + 2 ) + ( _ m + 2 ) 2 = 9 _ 6 m (m + 2) = m + 11 _ 6 m + 2 Square and simplify each expression. a) _ 2 ( z ) b) ( + _ 2 d 1 ) Substitute for a and _ m + 2 for b. Combine like terms. To solve the next two equations, we will have to square both sides of the equation twice to eliminate the radicals. Be very careful when you are squaring the binomials that contain a radical. EXAMPLE In-Class Example Solve each equation. a) c _c 9 = 1 b) 7k + 8 _ k + = 2 Answer: a) {2} b) {} Always use parentheses to organize your work when squaring both sides. Solve each equation. a) x + + x = b) w + 6 w + 1 = 1 a) This equation contains two radicals and a constant. Get one of the radicals on a side by itself, then square both sides. x + = x Subtract x from each side. ( x + ) 2 = ( x ) 2 Square both sides. x + = () 2 2()( x ) + ( x ) 2 Use the formula (a b) 2 = a 2 2ab + b 2. x + = 2 10 x + x Simplify. 602 CHAPTER 9 Radicals and Rational Exponents

77 These problems can be very long, so write out each step carefully and neatly to minimize mistakes. [ YOU TRY ] The equation still contains a radical. Therefore, repeat Steps 1. Begin by getting the radical on a side by itself. = 2 10 x Subtract x from each side. 20 = 10 x Subtract 2 from each side. 2 = x Divide by = ( x ) 2 Square both sides. = x Solve. The check is left to the student. The solution set is {}. b) Step 1: Get a radical on a side by itself. w + 6 w + 1 = 1 w + 6 = 1 + w + 1 Add w + 1 to each side. Step 2: Square both sides of the equation to eliminate a radical. ( w + 6 ) 2 = (1 + w + 1 ) 2 Square both sides. w + 6 = (1) 2 + 2(1)( w + 1 ) + ( w + 1 ) 2 Use the formula (a + b) 2 = a 2 + 2ab + b 2. w + 6 = w w + 1 Step : Combine like terms on the right side. w + 6 = w w + 1 Combine like terms. Step : The equation still contains a radical, so repeat Steps 1. Step 1: Get the radical on a side by itself. w + 6 = w w + 1 w + = 2 w + 1 Subtract w and subtract 2. We do not need to eliminate the 2 from in front of the radical before squaring both sides. The radical must not be a part of a sum or difference when we square. Step 2: Square both sides of the equation to eliminate the radical. (w + ) 2 = (2 w + 1 ) 2 Square both sides. w 2 + 8w + 16 = (w + 1) Square the binomial; 2 2 =. Steps and no longer apply. Step : Solve the equation. w 2 + 8w + 16 = 16w + Distribute. w 2 8w + 12 = 0 Subtract 16w and subtract. (w 2)(w 6) = 0 Factor. w 2 = 0 or w 6 = 0 Set each factor equal to zero. w = 2 or w = 6 Solve. Step 6: The check is left to the student. Verify that w = 2and w = 6each satisfy the original equation. The solution set is {2, 6}. Solve each equation. a) _ 2y + 1 y = 1 b) _ t + + t + 2 = 2 SECTION 9.7 Solving Radical Equations 60

78 BE CAREFUL Watch out for two common mistakes that students make when solving an equation like the one in Example b. 1) Do not square both sides before getting a radical on a side by itself. This is incorrect: ( w + 6 w + 1 ) 2 = 1 2 w + 6 (w + 1) = 1 2) The second time we perform Step 2, watch out for this common error. This is incorrect: (w + ) 2 = (2 w + 1 ) 2 w = 2(w + 1) On the left we must multiply using FOIL or the formula (a + b) 2 = a 2 + 2ab + b 2, and on the right we must remember to square the 2. Solve an Equation Containing a Cube Root We can solve many equations containing cube roots the same way we solve equations containing square roots except, to eliminate a cube root, we cube both sides of the equation. EXAMPLE 6 In-Class Example 6 Solve 2 _ k + 2 k + = 0. Answer: {} Write a procedure to help you solve equations containing a cube root. Solve 7a _ a 1 = 0. Begin by getting a radical on a side by itself. 7a + 1 = 2 _ a 1 Add 2 a 1 to each side. ( 7a + 1 ) = (2 _ a 1 ) Cube both sides to eliminate the radicals. 7a + 1 = 8(a 1) Simplify; 2 = 8. 7a + 1 = 8a 8 Distribute. 9 = a Subtract 7a; add 8. Check a = 9in the original equation. 7a _ a 1 = 0 _ 7(9) _ 9 1 =? 0 _ =? 0 2(2) =? 0 = 0 The solution set is {9}. [ YOU TRY 6 ] Solve _ r r + 2 = 0. Using Technology We can use a graphing calculator to solve a radical equation in one variable. First subtract every term on the right side of the equation from both sides of the equation, and enter the result in Y 1. Graph the equation in Y 1. The zeros or x-intercepts of the graph are the solutions to the equation. 60 CHAPTER 9 Radicals and Rational Exponents

79 We will solve x + = 2using a graphing calculator. 1) Enter x + 2 in Y 1. 2) Press ZOOM 6 to graph the function in Y 1 as shown. ) Press 2nd TRACE 2:zero, move the cursor to the left of the zero and press ENTER, move the cursor to the right of the zero and press ENTER, and move the cursor close to the zero and press ENTER to display the zero. The solution to the equation is x = 1. Solve each equation using a graphing calculator. 1) x 2 = 1 2) _ x 2 = ) _ x 2 = ) _ x = x + ) _ 2x 7 = x 1 6) x + 2 _ x 1 = 1 ANSWERS TO [YOU TRY] EXERCISES 1) a) {} b) 2) a) {, 2} b) {7} ) {} ) a) z _ 8 z + 16 b) d _ 10 d 1 ) a) {0, } b) { 1} 6) {} ANSWERS TO TECHNOLOGY EXERCISES 1) {} 2) {9} ) {9} ) {} ) {} 6) {} 9.7 Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective 1: Understand the Steps for Solving a Radical Equation 1) Why is it necessary to check the proposed solutions to a radical equation in the original equation? Sometimes there are extraneous solutions. 2) How do you know, without actually solving and checking the solution, that y = has no solution? The principle square root of a number cannot equal a negative number. Objective 2: Solve an Equation Containing One Square Root Fill in the blank with always, sometimes, or never to make the statement true. ) A negative number can sometimes be a solution to an equation containing a square root expression. ) An equation containing a square root expression will sometimes have extraneous solutions. Solve. ) q = 7 {9} 6) _ z = 10 {100} 7) w 2 = 0 { 9 } 8) _ r = 0 { _ 9 2 } 9) a + = 10) k + 8 = 2 11) _ b 11 = 0 {20} 12) d + = 0 {22} 1) _ g = 1 1) _ v = 6 1) _ f = 11 16) _ { 2 u + 12 = 17 } { _ 17) m = m 2 m + 6 {2} _ 29 } _ 18) b = b 2 + b 2 {6} 19) 9r 2 2r + 10 = r {} _ 20) p 2 p + 6 = 2p {2} SECTION 9.7 Solving Radical Equations 60

80 Square each binomial, and simplify. 21) (n + ) 2 n n + 2 2) (c 6) 2 c 2 12c + 6 Solve. 22) (z ) 2 z 2 6z + 9 2) (2k + 1) 2 k 2 + k + 1 2) c 7 = _ 2c ) p + 6 = _ {12} 12 + p { } 27) 6 + c 2 + c 9 = c 28) + z 2 + z 8 = z 29) w _ 10w + 6 = 0) _ 8t + 9 = t {1, } 1) v = 8 + _ {0, 2} v + {} 2) k = + 10k + ) m + = {2} m {1, 16} ) b + = 6 b {1, 2} ) y y = { } 6) r r + 2 = 2 {1} 7) r 2 8r 19 = r 9 8) x 2 + x + = x + 8 {10} { } Objective : Solve an Equation Containing Two Square Roots Solve. 9) _ 1 h = _ 1 8h { } 0) _ 6a 2 _ a + = 0 {11} 1) _ x + 6 _ 2 9x 9 = 0 {10} 2) _ q + 11 = 2 8q + 2 {2} ) m = 7 {6} ) = p {8} ) _ 2 t + + t 6 = 0 6) 2w 1 + _ 2 w + = 0 Square each expression, and simplify. 7) 2 ( x + ) 8) 2 ( y 8) x + 10 x 9) (9 + 2 y 16 y 2 a + ) 0) ( p + ) ) _ a + + a p (2 n 1 + 7) 2 2) ( _ + + p 2 2k ) 12n + _ 28 n k _ 0 2k 2 Solve. ) _ 2y 1 = 2 + y {, 1} ) _ n + = _ 2n {0, } ) 1 + _ s 2 = _ 2s + {2} 6) p = 6p 11 {6} 7) a + 19 _ a + 12 = 1 { 1 } 8) _ 2u + _ u + 1 = 1 {} 9) _ k + 1 k 1 = 2 {1, } 60) _ z _ z + 1 = 1 {, 7} 61) _ x + = x 11 62) _ c 7 = _ c + 1 6) _ v + v 2 = {2, 11} 6) _ 2y + 1 y = 1 {0, } Objective : Solve an Equation Containing a Cube Root 6) How do you eliminate the radical from an equation like x = 2? Raise both sides of the equation to the third power. 66) Give a reason why h = has no extraneous solutions. When you solve the equation you get h = 27, and the cube root of 27 is. Solve. 67) y = {12} 68) c = {27} 69) m = { 6} 70) _ t = 2 { 8} 71) 2x + = 1 { 72) 2 } a = { 7} 7) 6j 2 = _ j 7 { 1} 7) _ w + = _ 2w 11 {1} 7) y 1 2y = 0 { 2} 76) 2 2b + _ b = 0 { } 77) 2n 2 = 7n + { 1 2, } 78) c 2 c + 11 = c { 2, 1 } Extension Solve. 79) p 1/2 = 6 {6} 80) 2 = t1/2 { 9 } 81) 7 = (2z ) 1/2 {26} 82) (k + 1) 1/2 = {} 8) (y + ) 1/ = {2} 8) = (a 2) 1/ { 12} 8) n + 7 = 2 {9} 86) x = 1 87) 1 + _ r = r + 7 {9} 88) _ m 1 = m m {} _ 89) y + y + = y + 2 { 1} 90) 2d d + 6 = d + 6 {10} Mixed Exercises: Objectives 2 and Solve for the indicated variable. _ 2E 91) v = _ m for E E = mv2 2 92) V = 00VP for P m P = mv 00 _ A 9) c = a 2 + b 2 for b 2 9) r = π for A b 2 = c 2 a 2 A = πr CHAPTER 9 Radicals and Rational Exponents

81 _ 9) T = E σ for σ σ = E 96) r = _ V T π for V V = πr 97) The speed of sound is proportional to the square root of the air temperature in still air. The speed of sound is given by the formula. V S = 20 T + 27 where V S is the speed of sound in meters/second and T is the temperature of the air in Celsius. a) What is the speed of sound when the temperature is 17 C (about 1 F)? 20 m/sec b) What is the speed of sound when the temperature is 16 C (about 61 F)? 0 m/sec c) What happens to the speed of sound as the temperature increases? The speed of sound increases. d) Solve the equation for T. T = V s ) If the area of a square is A and each side has length l, then the length of a side is given by l = A A square rug has an area of 2 ft 2. a) Find the dimensions of the rug. ft ft b) Solve the equation for A. A = l 2 99) Let V represent the volume of a cylinder, h represent its height, and r represent its radius. V, h, and r are related according to the formula _ r = _ V πh a) A cylindrical soup can has a volume of 28π in. It is 7 in. high. What is the radius of the can? 2 in. b) Solve the equation for V. V = πr 2 h 100) For shallow water waves, the wave velocity is given by c = _ gh Kenna Love/Alamy where g is the acceleration due to gravity (2 ft/sec 2 ) and H is the depth of the water (in feet). a) Find the velocity of a wave in 8 ft of water. b) Solve the equation for H. H = c2 g 16 ft/sec 101) Refer to the formula given in Exercise 100. The catastrophic Indian Ocean tsunami that hit Banda Aceh, Sumatra, Indonesia, on December 26, 200, was caused by an earthquake whose epicenter was off the coast of northern Sumatra. The tsunami originated in about 1,00 ft of water. a) Find the velocity of the wave near the epicenter, in miles per hour. Round the answer to the nearest unit. (Hint: 1 mile = 280 ft.) 6 mph b) Banda Aceh, the area hardest hit by the tsunami, was about 60 mi from the tsunami s origin. Approximately how many minutes after the earthquake occurred did the tsunami hit Banda Aceh? (Exploring Geology, McGraw-Hill, 2008.) about 8 min. 102) The radius r of a cone with height _ h V and volume V is given by r = _ πh. A hanging glass vase in the shape of a cone is 8 in. tall, and the radius of the top of the cone is 2 in. How much water will the vase hold? Give an exact answer and an approximation to the tenths place. _ 2 π in ;. in Use the following information for Exercises 10 and 10. The distance a person can see to the horizon is approximated by the function D(h) = 1.2 h, where D is the number of miles a person can see to the horizon from a height of h ft. 10) Sig is the captain of an Alaskan crab fishing boat and can see.8 mi to the horizon when he is sailing his ship. Find his height above the sea. 16 ft 10) Phil is standing on the deck of a boat and can see.6 mi to the horizon. What is his height above the water? 9 ft Use the following information for Exercises 10 and 106. When the air temperature is 0 F, the wind chill temperature, W, in degrees Fahrenheit is a function of the velocity of the wind, V, in miles per hour and is given by the formula W(V) =.7.7V /2 10) Calculate the wind speed when the wind chill temperature is 10 F. Round to the nearest whole number mph 106) Find V so that W(V) = 20. Round to the nearest whole number. Explain your result in the context of the problem. ( When the wind chill temperature is 20 F, the speed of the wind is 16 mph. SECTION 9.7 Solving Radical Equations 607

82 R1) How has a good understanding of multiplying binomials helped you complete these exercises? R2) Look at Exercise 7. Next to your work solving this equation, explain, in words, the steps you used to solve the equation. 9.8 Complex Numbers What are your objectives for Section 9.8? 1 Find the Square Root of a Negative Number 2 Multiply and Divide Square Roots Containing Negative Numbers Add and Subtract Complex Numbers Multiply Complex Numbers Multiply a Complex Number by Its Conjugate 6 Divide Complex Numbers 7 Simplify Powers of i How can you accomplish each objective? Understand the definition of the imaginary number, i. Learn the definition of a complex number. Understand the property of Real Numbers and Complex Numbers. Complete the given example on your own. Complete You Try 1. Follow the explanation and example to create a procedure for this objective. Complete the given example on your own. Complete You Try 2. Write the procedure for Adding and Subtracting Complex Numbers in your own words. Complete the given example on your own. Complete You Try. Compare the steps for multiplying complex numbers to the notes you took on multiplying polynomials. Complete the given example on your own. Complete You Try. Understand the definition of a complex conjugate. Follow the explanation, and write the summary of Complex Conjugates in your notes. Complete the given example on your own. Complete You Try. Write the procedure for Dividing Complex Numbers in your own words. Complete the given example on your own. Complete You Try 6. Follow the explanation, and summarize how to simplify powers of i in your notes. Complete the given example on you own. Complete You Try CHAPTER 9 Radicals and Rational Exponents

83 Read the explanations, follow the examples, take notes, and complete the You Trys. 1 Find the Square Root of a Negative Number We have seen throughout this chapter that the square root of a negative number does not exist in the real number system because there is no real number that, when squared, will result in a negative number. For example, _ is not a real number because there is no real number whose square is. The square roots of negative numbers do exist, however, under another system of numbers called complex numbers. Before we define a complex number, we must define the number i. The number i is called an imaginary number. Definition The imaginary number i is defined as Therefore, squaring both sides gives us i = _ 1. i 2 = 1. Note i = _ 1 and i 2 = 1are two very important facts to remember. We will be using them often! Definition A complex number is a number of the form a + bi, where a and b are real numbers; a is called the real part, and b is called the imaginary part. The following table lists some examples of complex numbers and their real and imaginary parts. Complex Number Real Part Imaginary Part + 2i 2 1 7i 1 7 8i Note The complex number 8i can be written in the form a + bi as 0 + 8i. Likewise, besides being a real number, is a complex number because it can be written as + 0i. SECTION 9.8 Complex Numbers 609

84 Because all real numbers, a, can be written in the form a + 0i,all real numbers are also complex numbers. Property Real Numbers and Complex Numbers The set of real numbers is a subset of the set of complex numbers. Since we defined i as i = _ 1,we can now evaluate square roots of negative numbers. EXAMPLE 1 In-Class Example 1 Simplify. a) _ b) 11 c) Answer: a) 2i b) _ i 11 c) i 6 Create a procedure for simplifying square roots of negative numbers. Simplify. a) _ 9 b) _ 7 c) 12 a) _ 9 = 1 9 = _ 1 b) _ 7 = 1 7 = _ 9 = i = i 1 7 = i 7 12 = i c) 12 = _ 1 12 = _ 1 _ = i 2 = 2i Note In Example 1b) we wrote i 7 instead of 7 i,and in Example 1c) we wrote 2i instead of 2 i.we do this to be clear that the i is not under the radical. It is good practice to write the i before the radical. [ YOU TRY 1 ] Simplify. a) 6 b) 1 c) 20 2 Multiply and Divide Square Roots Containing Negative Numbers When multiplying or dividing square roots with negative radicands, write each radical in terms of i first. Remember, also, that since i = _ 1 it follows that i 2 = 1.We must keep this in mind when simplifying expressions. Note Whenever an i 2 appears in an expression, replace it with 1. EXAMPLE 2 In-Class Example 2 Multiply and simplify. 12 _ Answer: 6 Multiply and simplify. _ 8 _ 2 _ 8 _ 2 = i 8 i 2 = i 2 _ 16 Multiply. = ( 1)() Replace i 2 with 1. = Write each radical in terms of i before multiplying. 610 CHAPTER 9 Radicals and Rational Exponents

85 [ YOU TRY 2 ] Perform the operation, and simplify. a) _ 6 _ 72 b) 2 Add and Subtract Complex Numbers Just as we can add, subtract, multiply, and divide real numbers, we can perform all of these operations with complex numbers. Procedure Adding and Subtracting Complex Numbers 1) To add complex numbers, add the real parts and add the imaginary parts. 2) To subtract complex numbers, apply the distributive property and combine the real parts and combine the imaginary parts. EXAMPLE In-Class Example Add or subtract. a) (7 + i ) + ( + 6i ) b) ( i ) (10 + 2i ) Answer: a) i b) 6 i [ YOU TRY ] Add or subtract. a) (8 + i) + ( + 2i) b) (7 + i) ( i) a) (8 + i) + ( + 2i) = (8 + ) + ( + 2)i Add real parts; add imaginary parts. = 12 + i b) (7 + i) ( i) = 7 + i + i Distributive property = (7 ) + (1 + )i Add real parts; add imaginary parts. = + i Add or subtract. a) ( i) + (1 + 8i) b) (2 i) ( 1 + 6i) Multiply Complex Numbers We multiply complex numbers just like we would multiply polynomials. There may be an additional step, however. Remember to replace i 2 with 1. EXAMPLE In-Class Example Multiply and simplify. a) 7( 11i) b) (9 i )( 2 + i ) c) (9 + i )(9 i ) Answer: a) 28 77i b) + 7i c) 82 Multiply and simplify. a) ( 2 + i) b) (8 + i)( 1 + i) c) (6 + 2i)(6 2i) a) ( 2 + i) = i Distributive property SECTION 9.8 Complex Numbers 611

86 How does this compare to the techniques you learned in Section 6.? b) Look carefully at (8 + i)( 1 + i).each complex number has two terms, similar to, say, (x + )(x + ).How can we multiply these two binomials? We can use FOIL. F O I L (8 + i)( 1 + i) = (8)( 1) + (8)(i) + (i)( 1) + (i)(i) = 8 + 2i i + 12i 2 = i + 12( 1) Replace i 2 with 1. = i 12 = i c) Use FOIL to find the product (6 + 2i)(6 2i). F O I L 6 + 2i)(6 2i) = (6)(6) + (6)( 2i) + (2i)(6) + (2i)( 2i) = 6 12i + 12i i 2 = 6 ( 1) Replace i 2 with 1. = 6 + = 0 [ YOU TRY ] Multiply and simplify. a) (6 7i) b) ( + i)( + 8i) c) ( 2 9i)( 2 + 9i) Multiply a Complex Number by Its Conjugate In Section 9.6, we learned about conjugates of radical expressions. For example, the conjugate of + is. The complex numbers in Example c, 6 + 2i and 6 2i, are complex conjugates. Definition The conjugate of a + biis a bi. We found that (6 + 2i)(6 2i) = 0which is a real number. The product of a complex number and its conjugate is always a real number, as illustrated next. F O I L (a + bi)(a bi) = (a)(a) + (a)( bi) + (bi)(a) + (bi)( bi) = a 2 abi + abi b 2 i 2 = a 2 b 2 ( 1) Replace i 2 with 1. = a 2 + b 2 We can summarize these facts about complex numbers and their conjugates as follows: Summary Complex Conjugates 1) The conjugate of a + bi is a bi. 2) The product of a + bi and a bi is a real number. ) We can find the product (a + bi)(a bi) by using FOIL or by using (a + bi)(a bi) = a 2 + b CHAPTER 9 Radicals and Rational Exponents

87 EXAMPLE In-Class Example Multiply 7 + 2i by its conjugate using the formula (a + bi)(a bi) = a 2 + b 2. Answer: Always use parentheses! Multiply + iby its conjugate using the formula (a + bi)(a bi) = a 2 + b 2. The conjugate of + i is i. ( + i)( i) = ( ) 2 + () 2 a =, b = = = 2 [ YOU TRY ] Multiply 2 9iby its conjugate using the formula (a + bi)(a bi) = a2 + b 2. 6 Divide Complex Numbers 2 To rationalize the denominator of a radical expression like _ +, we multiply the numerator and denominator by,the conjugate of the denominator. We divide complex numbers in the same way. Procedure Dividing Complex Numbers To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Write the quotient in the form a + bi. EXAMPLE 6 In-Class Example 6 Divide. Write the result in the form a + bi. 9 a) 6 + i b) + i 11i Answer: a) 6 i 2 b) _ 17 _ i Be sure to use parentheses so that you distribute correctly. Divide. Write the quotient in the form a + bi. a) _ i b) a) b) 6 2i 7 + i _ i = ( + i) ( i) ( + i) Multiply the numerator and denominator by the conjugate of the denominator i = () 2 + () 2 numerators. Multiply (a + bi)(a bi) = a 2 + b i = _ i = _ 1 = _ _ 1 1 i Write the quotient in the form a + bi. Recall that we can find the product ( i)( + i)using FOIL or by using the formula (a + bi)(a bi) = a 2 + b 2. _ 6 2i 7 + i = (6 2i) ( 7 i) ( 7 + i) ( 7 i) Multiply the numerator and denominator by the conjugate of the denominator. = 2 6i + 1i + 2i2 Multiply using FOIL. ( 7) 2 + (1) 2 (a + bi)(a bi) = a 2 + b i 2 = _ + 8i = _ = _ _ 8 0 i = _ _ 2 i SECTION 9.8 Complex Numbers 61

88 [ YOU TRY 6 ] Divide. Write the result in the form a + bi. a) i b) + i _ 6 i 7 Simplify Powers of i All powers of i larger than i 1 (or just i) can be simplified. We use the fact that i 2 = 1to simplify powers of i. Let s write i through i in their simplest forms. i is in simplest form. i 2 = 1 i = i 2 i = 1 i = i i = (i 2 ) 2 = ( 1) 2 = 1 Let s continue by simplifying i and i 6. i = i i i 6 = (i 2 ) = (i 2 ) 2 i = ( 1) = ( 1) 2 i = 1 = 1i = i The pattern repeats so that all powers of i can be simplified to i, 1, i, or 1. EXAMPLE 7 In-Class Example 7 Simplify each power of i. a) i 12 b) i 22 c) i 1 d) i 1 Answer: a) 1 b) 1 c) i d) i [ YOU TRY 7 ] Simplify each power of i. a) i 8 b) i 1 c) i 11 d) i 7 a) Use the power rule for exponents to simplify i 8. Since the exponent is even, we can rewrite it in terms of i 2. i 8 = (i 2 ) Power rule = ( 1) i 2 = 1 = 1 Simplify. c) The exponent of i 11 is odd, so first use the product rule to write i 11 as a product of i and i 11 1 or i 10. i 11 = i 10 i Product rule = (i 2 ) i 10 is even; write i 10 in terms of i 2. = ( 1) i i 2 = 1 = 1 i Simplify. = i Multiply. Simplify each power of i. a) i 18 b) i 2 c) i 7 d) i 2 b) As in Example 7a), the exponent is even. Rewrite i 1 in terms of i 2. i 1 = (i 2 ) 7 Power rule = ( 1) 7 i 2 = 1 = 1 Simplify. d) The exponent of i 7 is odd. Use the product rule to write i 7 as a product of i and i 7 1 or i 6. i 7 = i 6 i Product rule = (i 2 ) 18 i 6 is even; write i 6 in terms of i 2. = ( 1) 18 i 2 = 1 = 1 i Simplify. = i Multiply. 61 CHAPTER 9 Radicals and Rational Exponents

89 Using Technology We can use a graphing calculator to perform operations on complex numbers or to evaluate square roots of negative numbers. If the calculator is in the default REAL mode the result is an error message ERR: NONREAL ANS, which indicates that _ is a complex number rather than a real number. Before evaluating _ on the home screen of your calculator, check the mode by pressing MODE and looking at row 7. Change the mode to complex numbers by selecting a + bi, as shown at the left below. Now evaluating _ on the home screen results in the correct answer 2i, as shown on the right below. Operations can be performed on complex numbers with the calculator in either REAL or a + bimode. Simply use the arithmetic operators on the right column on your calculator. To enter the imaginary number i, press 2 nd.. To add 2 iand + i, enter (2 i) + ( + i)on the home screen and press ENTER as shown on the left screen below. To subtract 8 + 6ifrom 7 2i, enter (7 2i) (8 + 6i)on the home screen and press ENTER as shown. To multiply iand 7 + i, enter ( i) (7 + i)on the home screen and press ENTER as shown on the middle screen below. To divide 2 + 9iby 2 i, enter (2 + 9i)/(2 i)on the home screen and press ENTER as shown. To raise ito the fifth power, enter ( i)^on the home screen and press ENTER as shown. 1 Consider the quotient ( + i)/( 7i). The exact answer is _ 6 + _ 7 i. The 6 calculator automatically displays the decimal result. Press MATH 1 ENTER to convert the decimal result to the exact fractional result, as shown on the right screen below. Perform the indicated operation using a graphing calculator. 1) Simplify 6 2) ( + 7i) + ( 8i) ) (10 i) ( + 8i) ) ( + 2i)(6 i) ) ( + i) (1 i) 6) ( i) ANSWERS TO [YOU TRY] EXERCISES 1) a) 6i b) _ i 1 c) 2i 2) a) 2 b) 6 ) a) 9 + 1i b) 11i ) a) i b) 12 + ic) 8 ) 8 6) a) _ 18 9 i b) _ _ 1 26 i 7) a) 1 b) 1 c) i d) i SECTION 9.8 Complex Numbers 61

90 ANSWERS TO TECHNOLOGY EXERCISES 1) 6i 2) 8 i ) 6 11i ) 2 + i ) i 6) i Exercises Do the exercises, and check your work. *Additional answers can be found in the Answers to Exercises appendix. Objective 1: Find the Square Root of a Negative Number Determine if each statement is true or false. 1) Every complex number is a real number. False 2) Every real number is a complex number. True ) Since i = _ 1, it follows that i 2 = 1. True ) In the complex number 6 + i, 6 is the real part and i is the imaginary part. False Simplify. ) 81 9i 6) 16 i 7) 2 i 8) _ 169 1i 9) _ 6 i 6 11) 27 i 10) 0 _ i 0 12) 7 i 1) 60 _ 2i 1 1) 28 2i 7 Objective 2: Multiply and Divide Square Roots Containing Negative Numbers Find the error in each of the following exercises, then find the correct answer. 1) _ 10 = _ ( 10) = _ 0 = _ 2 = 16) ( 7 ) 2 2 = ( 7) = _ 9 = Perform the indicated operation, and simplify. 17) _ 1 _ 19) 20 _ 10 20) 12 _ 6 21) ) 2) ( 1 ) ) _ ) ( _ 1 ) 2 Mixed Exercises: Objectives 6 2) Explain how to add complex numbers. Add the real parts, and add the imaginary parts. 26) How is multiplying (1 + i)(2 7i) similar to multiplying (x + )(2x 7)? Both are products of binomials, so we can multiply both using FOIL ) When i 2 appears in an expression, it should be replaced with what? 1 28) Explain how to divide complex numbers. Multiply the numerator and denominator by the conjugate of the denominator. Objective : Add and Subtract Complex Numbers Perform the indicated operations. 29) ( + 9i) + (7 + 2i) 0) (6 + i) + (8 i) + 11i 1 i 1) (1 8i) (9 + i) 2) ( 12 + i) ( 7 6i) 9i + 9i ) ( 1 6 i ) ( i ) 1 6 i ) ( i ) ( i ) ) 16i ( + 10i) + ( + i) 7i 8 + _ i 6) ( 6 i) + (2 + 6i) ( + i) 0 Objective : Multiply Complex Numbers Multiply and simplify. 7) (8 i) 2 1i 8) 6(8 i) 8 + 6i 9) 2 ( 9 + 2i) 6 + i 0) 1 2 (18 + 7i) i 1) i(6 + 11i) 2i 2) 6i( + 6i) 6 + 0i ) (2 + i)(1 + 6i) ) (2 + i)(10 + i) i i ) ( 1 + i)( 6i) 6) ( 9i)( i) i 21 2i 7) ( i)(9 i) 6 2i 8) ( i)(6 + 7i) 6 i 9) ( + i )( i ) 1 0) ( i )( + 2 i ) Objective : Multiply a Complex Number by Its Conjugate Identify the conjugate of each complex number, then multiply the number and its conjugate. 1) 11 + i 2) 1 2i conjugate: 11 i; product: 17 conjugate: 1 + 2i; product: ) 7i ) + 9i conjugate: + 7i; product: 8 conjugate: 9i; product: 97 ) 6 + i 6) 6 i conjugate: 6 i; product: 2 conjugate: 6 + i; product: CHAPTER 9 Radicals and Rational Exponents

91 7) How are conjugates of complex numbers like conjugates of expressions containing real numbers and radicals? Answers may vary. 8) Is the product of two complex numbers always a complex number? Explain your answer. No. For example, a complex number times its conjugate is a real number. Objective 6: Divide Complex Numbers Divide. Write the result in the form a + bi. 9) 61) 6) 6) _ 2 i 8i + i 2i _ + 7i _ _ 12 _ _ 2 _ 8i 6 + 7i 1 i 60) _ i _ _ i 17 i 62) _ i 6 i _ 61 + _ 6 61 i _ 7 29 _ 29 i 6) 9i + 10i _ _ 27 8 _ 8 _ 9 29 i + 2i i 66) _ _ 22 i 17 + _ 17 i 67) _ 2 + i 6i _ _ 27 i 68) _ 1 + 6i i _ _ i 69) 9 i 16 + i 9i 70) + 16i i Objective 7: Simplify Powers of i Simplify each power of i. Fill It In Fill in the blanks with either the missing mathematical step or reason for the given step. 71) i 2 = (i 2 ) 12 Rewrite i 2 in terms of i 2 using the power rule. = ( 1) 12 i 2 = 1 = 1 Simplify. 72) i 1 = i 0 i Product rule = (i 2 ) 1 i Rewrite i 0 in terms of i 2 using the power rule. = ( 1) 1 i i 2 = 1 = 1 i Simplify. = i Multiply. 7) i 2 1 7) i ) i ) i ) i 9 i 78) i 19 i 79) i i 80) i 29 i 81) i 2 i 82) i 0 1 8) i 2 1 8) i i 8) (2i) 2i 86) (2i) ) ( i) ) ( i) 1 i Expand. 89) ( 2 + i) 12 6i 90) ( i) 117 i Simplify each expression. Write the result in the form a + bi. 91) 1 + _ i 2 92) 7 8 9) 8 8 i 9) i 9) i 2 96) i 2 Used in the field of electronics, the impedance, Z, is the total opposition to the current flow of an alternating current (AC) within an electronic component, circuit, or system. It is expressed as a complex number Z = R + Xj, where the i used to represent an imaginary number in most areas of mathematics is replaced by j in electronics. R represents the resistance of a substance, and X represents the reactance. The total impedance, Z, of components connected in series is the sum of the individual impedances of each component. Each exercise contains the impedance of individual circuits. Find the total impedance of a system formed by connecting the circuits in series by finding the sum of the individual impedances. 97) Z1 = + 2j Z 2 = 7 + j Z = j 99) Z1 = 2j Z 2 = j Z = 16 + j 98) Z1 = + j Z 2 = 9 + 6j Z = 1 + 9j 100) Z1 = 1.j Z 2 = + 0.j Z = 7 1.0j 7 i _ R1) Explain, in your own words, the meaning of the imaginary number i. How does this help us find the square root of a negative number? R2) Which learning objectives from previous chapters helped you master the concepts of this exercise set? SECTION 9.8 Complex Numbers 617