Section 6.2 PRE-ACTIVITY PREPARATION Exponents Exponents enable you to simplify the presentation of a numerical expression containing repeated multiplication into a concise form that is easier to read and use in practical applications. For instance, you may have already used exponents to present the prime factorization of a number. Exponents are often utilized in the analysis of statistical data, and they are used in science and engineering when presenting very large or very small numbers in scientific notation. In business applications, using exponents greatly simplifies the processes of calculating growth of savings or retirement of debt. Although exponents can be negative as well as positive numbers, this chapter will only address the use of positive exponents. LEARNING OBJECTIVES Write expressions in exponential notation. Evaluate expressions having positive exponents. Order a mixed set of signed numbers and numerical expressions. TERMINOLOGY PREVIOUSLY USED base evaluate exponent exponent form exponential notation expression power simplify term NEW TERMS TO LEARN cubed exponential expression raised to squared 569
570 Chapter 6 Signed Numbers, Exponents, and Order of Operations BUILDING MATHEMATICAL LANGUAGE The notation you can use to represent repeated multiplication of the same number is called exponential notation. exponent For example, 7 7 7 7 7 in exponent form is 7 5 seven to the fifth power or seven raised to the fifth power base The exponent 5 denotes five factors of the base number 7. For example, 3 3 3 5 5 written in exponential notation is 3 3 5 2 The base can be any kind of number a whole number, a fraction, a decimal number, positive or negative. Sometimes the base number might be written within parentheses. For example, 5 3 might also be written as (5) 3, or 0.06 2 might be written as (0.06) 2. Negative number bases and fraction bases, however, must be written in parentheses for clear presentation 2 2 of the base number: ( 3) 5 and, for example. 5 Any base number raised to the first power means that number is a factor one time. For example, 7 1 = 7, 23 1 =23, and so on. Therefore, it is customary to write 7 1 as 7, 23 1 as 23, and so on. A base number raised to the second power is commonly referred to as that number squared. For example, 7 7 = 7 2 seven squared A base number raised to the third power is often referred to as that number cubed. For example, 21 21 21 = 21 3 twenty-one cubed Any number to the power of zero is equal to 1. For example, 7 0 = 1, 23 0 =1, and so on. Terms written with exponents can be combined with other mathematical symbols to form new expressions, as in the following examples: 7 + 3 2 10 2 + 2 4
Section 6.2 Exponents 571 Expressions having exponents are sometimes referred to as exponential expressions. To evaluate or to simplify an exponential expression is to simplify the expression to one number. The most accurate way to simplify an exponential expression is to write out its factors and then multiply them. (5) 3 = 5 5 5 = 25 5 = 125 MODELS Model 1 Evaluate each of the following: A (0.06) 2 (0.06) 2 = 0.06 0.06 = 0.0036 Answer 006. 006. 0. 0036 B ( 3) 5 ( 3) 5 = 3 ( 3) ( 3) ( 3) ( 3) OR Use the shortcut for multiplying more than two signed numbers. With five = +9 ( 3) ( 3) ( 3) negative factors, the answer will be = 27 ( 3) ( 3) negative. Multiply 3 3 3 3 3 and = +81 ( 3) attach a negative sign: = 243 Answer: 243 C 2 5 2 2 2 2 2 5 = 5 5 = 4 25 Answer D 10 2 + 2 4 Evaluate each term separately: 10 2 = 10 10 = 100 2 4 = 2 2 2 2 = 16 10 2 + 2 4 = 100 + 16 = 116 Answer
572 Chapter 6 Signed Numbers, Exponents, and Order of Operations E Evaluate ( 4) 2 and 4 2. There is an important distinction to be made between ( 4) 2 and 4 2. ( 4) 2 Because of the parentheses around the 4, the term ( 4) 2 represents the base number 4 raised to the second power: ( 4) ( 4) = +16 That is, ( 4) 2 = +16 Answer 4 2 On the other hand, the negative sign in the expression 4 2, indicates the opposite of the base number 4 raised to the second power: 4 2 = the opposite of 4 2 = the opposite of 4 4 or the opposite of 16 That is, 4 2 = 16 Answer F Evaluate ( 4) 3 and 4 3. Note that, with an odd power, both simplified answers are the same. ( 4) 3 = ( 4) ( 4) ( 4) = 64 Answer 4 3 = the opposite of 4 4 4, or the opposite of 64, or 64 Answer Model 2 Combine your knowledge of signed numbers and exponents, and list the following sets of numerical expressions in order from lowest to highest value. A 2, 42, 12, 10, ( 5) 2 Evaluate the expressions with exponents: 4 2 = (4 4) = 16 ( 5) 2 = ( 5) ( 5) = +25 Original list: 2, 4 2, 12, 10, ( 5) 2 List after evaluating terms with exponents: 2, 16, 12, 10, 25 Visualize on a number line (optional). 16 10 2 12 25 16 < 10 < 2 < 12 < 25 That is, from lowest to highest: 4 2, 10, 2, 12, ( 5) 2 Answer
Section 6.2 Exponents 573 B 1 3 2,, 2 0, ( 2. 1 ) 2, 4. 2 Evaluate the expressions with exponents: Original list: 2 1 3 2 0 2 1 2,,, (. ), 4. 2 3 2 = 3 3 = 9 (2.1) 2 = 2.1 2.1 = 4.41 List after evaluating terms with exponents: 1 9,, 2 0, 4. 41, 4. 2 Since two are decimals, compare as decimals. 9, 0. 5, 0, 441., 42. Visualize on a number line (optional). 4.2 0.5 0 4.41 9 4.2 < 0.5 < 0 < 4.41 < 9 Using the original numbers, the list from lowest to highest is: -4.2, - 1 2 2 2, 0, ( 2.1 ), 3 Answer ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Misinterpreting the exponent as a factor Evaluate: 8 4 8 4 = 4 8 =32 Answer: 32 The exponent indicates how many times the base should be multiplied by itself. Write out the factors and count them to assure that you have written the base as a factor as many times as the exponent indicates. Evaluate: 8 4 8 4 = 8 8 8 8 = 64 8 8 = 512 8 = 4096
574 Chapter 6 Signed Numbers, Exponents, and Order of Operations PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with exponents how to write in exponential notation how to evaluate exponential expressions how to order a mixed set of numbers and numerical expressions
Section 6.2 ACTIVITY Exponents PERFORMANCE CRITERIA Writing expressions in exponent form correct notation Evaluating expressions with exponents accuracy Ordering a set of numbers and numerical expressions accurate conversions to the same form correct order CRITICAL THINKING QUESTIONS 1. What is the meaning of the exponent? 2. What kind of number(s) can you use for the base? 3. What does it mean to have an exponent of one (1)? 575
576 Chapter 6 Signed Numbers, Exponents, and Order of Operations 4. What is the value of one (1) raised to any power? 5. What is the result when you evaluate a negative number raised to an even number exponent? 6. What is the result when you evaluate a negative number raised to an odd number exponent? 7. How can you represent a number added to itself four times versus a number multiplied by itself four times?
Section 6.2 Exponents 577 TIPS FOR SUCCESS Write out the factors when evaluating an expression with exponents. Use the shortcut for determining the sign of the simplified answer when the base of the exponential expression is a negative number: An even exponent means an even number of negative factors, resulting in a positive answer. An odd exponent means an odd number of negative factors, resulting in a negative answer. When the base is a decimal number, pay careful attention to the placement of the decimal point in the simplified answer. DEMONSTRATE YOUR UNDERSTANDING 1. Write each expression in exponent form and then evaluate it. Expression Exponent Form Evaluation a) 7 7 3 3 3 b) 2 2 2 2 5 5 11 c) 3 3 + 5 5 5
578 Chapter 6 Signed Numbers, Exponents, and Order of Operations 2. Evaluate the following exponential expressions: a) 4 3 2 2 3 0 b) 3 1 5 c) 2 2 3 d) 11 2 e) 5 2 + 10 3 f) (0.03) 3 g) 20 2 h) 1 57 3. Which is greater? 2 3 or 3 2 4. Which is greater? 5 3 or 3 5
Section 6.2 Exponents 579 5. What number is 1 less than 10 3? 6. Order the following from lowest to highest value: 1, 5, 12, 10, 5 8 ( ) 2 2 IDENTIFY AND CORRECT THE ERRORS Identify the error(s) in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here? Identify the Errors Correct Process 1) Evaluate: 6 2 2 5 Used the exponent as a factor. 6 2 means 6 6, not 2 6. 2 5 means 2 2 2 2 2, not 5 2 6 2 2 5 = 6 6 2 2 2 2 2 4 4 2 = 36 32 = 1152 36 32 72 1080 1152 2) Evaluate: 1 4 2
580 Chapter 6 Signed Numbers, Exponents, and Order of Operations Worked Solution What is Wrong Here? 3) Evaluate: 9 2 Identify the Errors Correct Process 4) Evaluate: ( 0.02) 3 ADDITIONAL EXERCISES 1. Write in exponential notation: a) 9 9 2 2 2 b) 10 6 6 6 6 3 3 2. Evaluate each of the following: a) 7 3 b) ( 6) 2 c) d) (0.5) 3 e) 8 2 + 8 f) 3 2 5 2 1 4 g) 40 2 h) (2.1) 2 i) ( 9) 3 j) ( 1) 15 3. Order the following from lowest to highest value: 2 3, 4 2, 2 3, ( 4) 2, 2 5