Exponents and Real Numbers

Transcription

1 Exponents and Real Numbers MODULE? ESSENTIAL QUESTION What sets of numbers are included in the real numbers? CALIFORNIA COMMON CORE LESSON.1 Radicals and Rational Exponents N.RN.1, N.RN. LESSON. Real Numbers N.RN. Image Credits: photodisc/getty Images Real-World Video Zoo managers must determine the amount of food needed for a healthy diet for the animals. Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 5

2 Are YOU Ready? Complete these exercises to review skills you will need for this module. Exponents EXAMPLE Write (-5) as a multiplication of factors. (-5)(-5)(-5) Write the base -5 times itself times. Personal Math Trainer Online Practice and Help EXAMPLE Write using a base and an exponent. 1 6 The base is 1 and the exponent is the number of factors 6. Write each expression as a multiplication of factors (-6) Write each expression using a base and an exponent (-) (-) Evaluate Powers EXAMPLE Evaluate (-) 4-6. (-)(-)(-)(-) Write as a multiplication of factors. Simplify. Evaluate each expression (-9) (5) 0-5 Squares and Square Roots EXAMPLE Find _ Because 7 7 = 49, 7 is the square root of 49. Find each square root. 10. _ _ _ 10,000 6 Unit 1A

3 Reading Start-Up Visualize Vocabulary Fill in the missing information in the chart below. Word Definition Example exponent 4 = = 81 4 is the exponent. Vocabulary Review Words exponent (exponente) rational numbers (números racionales) irrational numbers (números irracionales) Preview Words radical expression radicand index real numbers closed rational numbers A number that can be written in the form b/a, where a and b are integers and b 0 A real number that cannot be expressed as the ratio of two integers π, _, e Understand Vocabulary To become familiar with some of the vocabulary terms in the module, complete the following sentences using appropriate preview words. You may refer to the module, the glossary, or a dictionary. 1. The consist of the rational numbers and the irrational numbers.. The of a radical expression indicates which root to take of the. Active Reading Key-Term Fold Before beginning the module, create a Key-Term Fold Note to help you organize what you learn. Write a vocabulary term on each tab of the key-term fold. Under each tab, write the definition of the term and an example of the term. Module 7

4 GETTING READY FOR Exponents and Real Numbers Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module. N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Key Vocabulary radical (radical) An indicated root of a quantity. What It Means to You You can rewrite expressions containing radicals as expressions with rational exponents and vice versa. EXAMPLE N.RN.1 Rewrite 5 as a radical and simplify. Raising to the 5 power is the same as taking the 5th root. 5 = 5 _ = 5 _ 5 = N.RN. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Key Vocabulary exponent (exponente) The number that indicates how many times the base in a power is used as a factor. What It Means to You You can use the properties of exponents to rewrite and simplify radical expressions and expressions containing rational exponents. EXAMPLE N.RN. Simplify 8 5_. 8 5_ = 8 5 = ( 8 ) 5 = ( _ 8 ) 5 = ( _ ) 5 = 5 = Visit to see all CA Common Core Standards explained. 8 Unit 1A

5 ? LESSON.1 ESSENTIAL QUESTION Radicals and Rational Exponents How are radicals and rational exponents related? N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Also N.RN. EXPLORE ACTIVITY N.RN.1 Defining Rational Exponents The radical symbol _ 1 is used to indicate square roots. Roots other than square roots are indicated by using an index with the radical symbol. An expression that contains radicals is a radical expression. Index _ 15 Radicand Complete the steps below to explore the relationship between radical expressions and rational exponents. Recall that ( a m ) n = a mn. A _ 4 = 4k STEP 1 ( ) = ( ) Square both sides of the equation. STEP 41 = 4k Power of a Power Property STEP = If b m = b n, then m = n. STEP 4 = k Solve for k. STEP 5 Substitute your value for k in the original equation: _ 4 = 4 B _ 8 = 8k STEP 1 ( ) STEP 81 = 8k = ( ) Cube both sides of the equation. Power of a Power Property STEP = If b m = b n, then m = n. STEP 4 = k Solve for k. STEP 5 Substitute your value for k in the original equation: _ 8 = 8 REFLECT 1. Make a Conjecture How can you write the square root or the cube root of a number n using an exponent? Lesson.1 9

6 Simplifying Expressions with Rational Exponents Math On the Spot Definition of b _ n General Rule Examples A number raised to the power of n is equal to b = _ b the nth root of that number: 5 = _ 5 = 5 b n = _ n b,b = _ b 8 = _ 8 = where b 0 and n is an integer > 1.b 4 4 = _ b = _ 81 = You can use the definition of b n to simplify expressions with rational exponents. EXAMPLE 1 N.RN.1, N.RN. Simplify each expression. A = _ 64 = _ 4 Use the definition of b _ n. Rewrite the radicand as a cube. = 4 B = _ - _ 81 Use the definition of b _ n. = 5 _ 5 - _ 9 Rewrite both radicands as powers. = - 9 = -7 REFLECT. Justify Reasoning Is b n where b > 0 and n is a positive integer always a positive integer? Justify your answer with reasoning or a counterexample. YOUR TURN Simplify each expression Personal Math Trainer Online Practice and Help Unit 1A

7 Using Properties with Rational Exponents You can use properties of exponents to simplify expressions containing rational exponents. Recall the following properties of exponents. Properties of Exponents Property Numerical example Math On the Spot Product of Powers Propertya m a n = a m + n = 5 = Quotient of Powers Property a m = a m - n, a 0 5 a = n = 4 Power of a Product Property (a b) n = a n b n ( ) = = 6 Power of a Quotient Property ( a b ) n = a n n, b 0 ( ) = 1 b = 4 Power of a Power Property ( am ) n = a mn ( ) = = 4 = 16 Negative Exponent Propertya -n = _ a n, a 0 - = _ = 4 To simplify expressions with rational exponents use the properties of exponents and the definition of b _ n. EXAMPLE N.RN.1, N.RN. Simplify each expression. A 15 _ B 81 _ 4 15 _ = 15 Write the exponent as a product = ( 15 ) Power of a Power Property = ( _ 15 ) _ Definition of b n = ( _ 5 ) 15 = 5 = (5) = 5 81 _ 4 = 81 4 Write the exponent as a product = ( 81 4 ) Power of a Power Property = ( _ 4 81 ) _ Definition of b n = ( 4 _ 4 ) 81 = 4 = () = 7 Math Talk Mathematical Practices Why is it better to evaluate the radical expression first before raising to a power? Lesson.1 1

8 My Notes C 64 _ = 64 Write the exponent as a product = ( 64 ) Power of a Power Property = ( _ 64 ) _ Definition of b n = ( _ 8 ) 64 = 8 = 8 = 51 REFLECT 7. Communicate Mathematical Ideas Explain how you simplify any expression of the form a m n. 8. Example A showed how to simplify 15. Is it possible to simplify 15 using similar steps? Explain. Personal Math Trainer Online Practice and Help YOUR TURN Simplify each expression _ 4_ Rational Exponents in Real-World Contexts You can use rational exponents to describe relationships in real-world contexts, such as the relationship between an animal s required caloric intake and its mass. Math On the Spot Unit 1A

9 EXAMPLE N.RN.1, N.RN. The approximate number of Calories C that an animal needs each day is given by C = 7 m _ 4, where m is the animal s mass in kilograms. Find the number of Calories that each animal needs daily. A a Siberian tiger with a mass of 56 kilograms C = 7 m _ 4 = 7( 56) _ 4 Substitute 56 for m. = 7 (56) 4 4 = _ 4 = 7 ( 56 4 ) Power of a Power Property = 7 ( 4 _ 56 ) Definition of b _ n = 7 ( 4 _ 4 4 ) 56 = 4 4 = 7 (4) = 7 64 = 4608 The tiger needs 4608 Calories per day. Image Credits: Digital Vision/Getty Images B an Asian elephant with a mass of 4096 kilograms C = 7 m _ 4 = 7( 4096) _ 4 Substitute 4096 for m. = 7 (4096) 4 4 = _ 4 = 7 ( ) Power of a Power Property = 7 ( 4 _ 4096 ) Definition of b _ n = 7 ( 4 _ 8 4 ) 4096 = 8 4 = 7 (8) = 7 51 = 6,864 The elephant needs 6,864 Calories per day. YOUR TURN 1. Use C = 7 m _ 4 to find the number of Calories that an Australian shepherd dog with a mass of 16 kilograms needs each day. Personal Math Trainer Online Practice and Help Lesson.1

10 Guided Practice Simplify each expression. (Example 1) Simplify each expression. (Example ) _ 4_ 5_ _ Near Earth s surface, the time t required for an object to fall a distance d is given by t = 4 d, where t is measured in seconds and d is measured in feet. Find the time it will take an object to fall 100 feet. (Example ) 14. The relationship between the radius, r, of a sphere and its volume, V, is r = ( V 4π ). What is the radius of a sphere that has a volume of 6π cubic units? (Example )? 15. The relationship between the radius, r, of a sphere and its surface area, A, is r = ( A 4π ). What is the radius of a sphere that has a surface area of 64π square units? (Example ) ESSENTIAL QUESTION CHECK-IN 16. What does the denominator of a rational exponent represent? 4 Unit 1A

11 Name Class Date.1 Independent Practice Simplify each expression _ , _ _ - 81 _ _ Use the equation t = 4 d, where t is the time in seconds and d is the distance in feet, to find the time it takes for an object to fall each distance feet. 6 feet feet 5. 1 foot N.RN.1, N.RN. 6. Use the equation t = to determine the height from which an object fell if it took 4 seconds to reach the ground. 4 d If a right triangle has legs of length a and b and hypotenuse of length c, then c = ( a + b ). Determine the length of the hypotenuse for a right triangle with the given leg lengths. 7. a = 5 in., b = 1 in. 8. a = 6 cm, b = 8 cm 9. a = 1 mi, b = 9 mi Write the name of the property that is demonstrated by each equation. 0. ( 5 ) = = 4. - = 9. ( _ ) = = ( ) = Personal Math Trainer Online Practice and Help For each property, give an example that demonstrates the property. Do not use an example that has been shown in this lesson. 6. Power of a Power 7. Power of a Product 8. Negative Exponent 9. Quotient of Powers 40. Power of a Quotient Lesson.1 5

12 FOCUS ON HIGHER ORDER THINKING Work Area 41. Communicate Mathematical Ideas Use the Commutative Property of Multiplication and the properties of rational exponents to rewrite.5 m n as two equivalent radical expressions. Explain what these two expressions mean about different ways to evaluate.5 m n. 4. Multiple Representations Show that a 6 can be written as a perfect square and as a perfect cube. 4. Critical Thinking Use the Commutative Property of Multiplication and the Associative Property of Multiplication to show the Power of a Product Property (a b ) n = a n b n is true. 44. Critique Reasoning Jay said that by the Quotient of Powers property, 0 5 = 0 5 = 0 = 0. Is this correct? Explain. 0 6 Unit 1A

13 ? LESSON. Real Numbers ESSENTIAL QUESTION What are the subsets and properties of real numbers? N.RN. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. EXPLORE ACTIVITY 1 Prep for N.RN. Understanding Real Numbers Recall that a rational number can be expressed in the form p q, where p and q are integers and q 0. The decimal form of a rational number either terminates or repeats. For instance, _ 4 = 0.75 and _ 5 6 = 0.8. An irrational number cannot be written as the quotient of two integers, and its decimal form is nonrepeating and nonterminating. Examples of irrational numbers include square roots of non-perfect squares and cube roots of non-perfect cubes. For instance, the decimal form of _ is , which neither repeats nor terminates. Real numbers consist of all rational and irrational numbers. The Venn diagram shows subsets of the set of real numbers. Each of the subsets includes one example of a real number belonging to that subset. Classify each number by writing it in the most specific area of the diagram. A -5 Real Numbers B 0 C - _ 10 D E 8 F -1 G _ Rational Numbers Integers - Whole Numbers 154 Irrational Numbers REFLECT 1. Communicate Mathematical Ideas How does the diagram show that all whole numbers are integers and that all integers are rational numbers? π Animated Math Lesson. 7

14 Properties of Real Numbers All real numbers have the following properties with respect to addition and multiplication. Math On the Spot Properties of Real Numbers Commutative Property of Addition a + b = b + a Associative Property of Addition (a + b) + c = a + (b + c) Additive Identity The additive identity is 0, because a + 0 = a. Additive Inverse The additive inverse of a is a, because a + ( a) = 0. Commutative Property of Multiplication a b = b a Associative Property of Multiplication (a b) c = a (b c) Multiplicative Identity Multiplicative Inverse The multiplicative identity is 1, because 1(a) = a. The multiplicative inverse of a for a 0 is a, because a ( a ) = 1. Distributive Property a(b + c) = ab + ac A set of numbers is closed under an operation if the result of the operation on any two numbers in the set is also a number in that set. EXAMPLE 1 N.RN. A Determine whether the set { 1, 0, 1} is closed under addition. My Notes Add each pair of elements in the set. Check whether each sum is in the set. 1 + ( 1) = [ ] = = 0 B = = = [ ] The sums and are not in the original set, so the set { 1, 0, 1} is not closed under addition. Show that the set of irrational numbers is not closed under addition. Find two irrational numbers whose sum is not an irrational number. _ + (- _ ) = 0 0 is not irrational. The set of irrational numbers is not closed under addition. REFLECT. Give an example that shows the set of integers is not closed under division. 8 Unit 1A

15 . Give an example that shows the set of irrational numbers is not closed under multiplication. 4. Communicate Mathematical Ideas Under which operations is the set of whole numbers closed? Not closed? Explain. YOUR TURN 5. Is the set {, 0, } closed under addition? Explain. 6. Give an example that shows the set of irrational numbers is not closed under Personal Math Trainer Online Practice and Help division. EXPLORE ACTIVITY Proving that Sets are Closed N.RN. The set of integers is closed under addition and multiplication. Complete Steps 1 5 to prove that the set of rational numbers is closed under addition. Begin with the definition of rational numbers a and b. STEP 1 Let a and b be numbers. Then a = _ p q and b = _ r s, where p, q, r, and s are integers and q and s are not 0. STEP Find a common denominator. STEP Multiply. a + b = _ p q + _ r s = p _ q ( s _ s ) + r _ s ( q _ q ) = + STEP 4 Add the numerators. = Lesson. 9

16 EXPLORE ACTIVITY (cont d) STEP 5 Conclusion: Because the numerator pq + qr and the denominator qs are both, ps + qr qs is a rational number. Summary: For each set of numbers in rows A D, enter Yes if the set of numbers is closed under the operation. Enter No if the set of numbers is not closed under the operation. A B C D Closure of Number Sets Set Addition Subtraction Multiplication Division Real numbers Irrational numbers Rational numbers Integers REFLECT 7. Critique Reasoning In Step 5, how do you know that ps + qr and qs are integers? 8. Critique Reasoning How does a + b = numbers is closed under addition? ps + qr qs prove that the set of rational 9. Draw Conclusions Given that the set of rational numbers is closed under addition, how can you prove that the set of rational numbers is closed under subtraction? 40 Unit 1A

17 EXPLORE ACTIVITY N.RN. Complete Steps 1 8 to prove that the sum of an irrational number and a rational number is irrational. STEP 1 STEP STEP Let a be an irrational number and let b be a rational number. Then b = _ r s, where r and s are and s 0. The sum a + b must be either rational or irrational. Assume that the sum a + b is rational. Then a + b = and q 0. Subtract b from each side., where p and q are a + b - a + b = p _ q = p _ q - This proof is called a proof by contradiction. This is when an assumption is made at the beginning, and if the logical outcome is false, then the assumption must be false. STEP 4 STEP 5 Substitute _ r s for b. Find a common denominator. a = a = p _ q ( s _ s ) - r _ s ( q _ q ) STEP 6 Multiply. a = - STEP 7 Subtract. a = STEP 8 ps - qr qs Because ps qr and qs are integers with qs 0, is a rational number. But in Step 1, a is given as an irrational number. This means that the assumption that was incorrect. So the sum of an irrational number and a rational number must be. REFLECT 10. Justify Reasoning Use the results of Explore Activity to justify that the difference of an irrational number and a rational number is irrational. Math Talk Mathematical Practices How do you know that ps - qr and qs are integers? Lesson. 41

18 Guided Practice Tell whether each set is closed under the given operation. (Example 1) 1. {0, 1}; multiplication. {0, 1}; addition. even integers; addition 4. Prove that the set of rational numbers is closed under multiplication. (Explore Activity ) Let a and b be rational numbers. Then a = _ p q and b = _ r s, where p, q, r, and s are integers and q and s are not 0. a b = = a b = a b b = (b 0) b a = (b 0) Because pr and qs are integers, is a rational number. So the set of rational numbers is closed under multiplication. 5. Prove that the product of an irrational number and a nonzero rational number is irrational. (Explore Activity ) Let a be an irrational number and let b be a nonzero rational number. Then b = _ r s, where r and s are integers and r and s are not. The product a b must be either rational or irrational. Assume a = (s 0) a = (r 0) a =? that Then a b =, where p and q are integers and q is not 0. ESSENTIAL QUESTION CHECK-IN The final statement shows that a is a number. But a is given as an irrational number. Therefore the assumption is incorrect. So the product of an irrational number and a nonzero rational number is. 6. How do you show that a set of numbers is not closed under a given operation? 4 Unit 1A

19 Name Class Date. Independent Practice N.RN. Write two numbers that fit each description. If there is no such number, write none. 7. negative integer Personal Math Trainer Online Practice and Help Tell whether the set is closed under the operation. If it is not closed, justify your answer using an example. 16. negative integers; subtraction 8. negative rational number that is not an integer 17. negative integers; addition 9. irrational integer 18. rational numbers; division 10. negative irrational number 19. {, 0, }; multiplication Using whole, integer, rational, and irrational, name all the subsets of the real numbers to which each number belongs negative integers; multiplication 1. negative rational numbers; multiplication 1. π. positive irrational numbers; addition 1. - _ _ 6 _ 9 0 _ 7. positive irrational numbers; multiplication 4. {0, 1, 10}; multiplication 5. even integers; subtraction Lesson. 4

20 FOCUS ON HIGHER ORDER THINKING Work Area 6. Make a Conjecture Consider any subset of the real numbers that consists only of negative numbers. What can you conclude about whether the set is closed under multiplication? Explain. 7. Communicate Mathematical Ideas Explain why any real number must be either a rational number or an irrational number. 8. Explain the Error Drew wanted to determine whether the set of rational numbers is closed under division. He concluded that the set is not closed because _ 4 =, 4 and is an integer. Explain Drew s error. 9. Draw Conclusions Consider a set of numbers that is closed under addition and subtraction. What number must be in such a set? Explain. 0. Draw Conclusions Consider a set of numbers that is closed under multiplication and division. What number must be in such a set? Explain. 44 Unit 1A

21 Ready to Go On?.1 Radicals and Rational Exponents Simplify each expression _ _ 5 5_ _ Write the name of the property that is demonstrated by each equation = = 4 8 Personal Math Trainer Online Practice and Help 7. Use the equation t = a _ to find the value of t when a = 16.. Real Numbers Using whole, integer, rational, and irrational, name all the subsets of the real numbers to which each number belongs. 8. _ 5 9. _ _ 8 Tell whether the set is closed under the operation. If it is not closed, justify your answer using an example. 1. irrational numbers; addition 1. rational numbers; multiplication? ESSENTIAL QUESTION 14. What sets of numbers are included in the real numbers? Module Quiz 45

22 MODULE MIXED REVIEW Assessment Readiness Personal Math Trainer Online Practice and Help 1. Evaluate each expression. Is the expression equal to 4? Select Yes or No for expressions A C. _ A. 8 Yes No B. 16 Yes No C. 6 Yes No. Consider the set of real numbers. Choose True or False for each statement. A. The product of a nonzero rational number and an irrational number is always irrational. True False B. The product of two rational numbers is always rational. True False C. The product of two irrational numbers is always irrational. True False. Simplify the expression. Explain how the expression and its simplified form - show that the set of irrational numbers is not closed under division. 4. A recipe calls for 150 grams of cheddar cheese. Dylan has 4.0 ounces of cheddar cheese. How many more ounces will he need for the recipe? Explain how you solved this problem. Use 1 kilogram. pounds. 46 Unit 1A