THE POWER RULES. Raising an Exponential Expression to a Power


 Alvin Watts
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1 8 (5) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar of the Rules Applications In Section 5. ou learned some of the basic rules for working with eponents. All of the rules of eponents are designed to make it easier to work with eponential epressions. In this section we will etend our list of rules to include three new ones. Raising an Eponential Epression to a Power An epression such as ( ) consists of the eponential epression raised to the power. We can use known rules to simplif this epression. ( ) Eponent indicates two factors of. Product rule: Note that the eponent is the product of the eponents and. This eample illustrates the power of a power rule. Power of a Power Rule If m and n are an integers and a 0, then (a m ) n a mn. E X A M P L E A graphing cannot prove that the power of a power rule is correct, but it can provide numerical support for it. Using the power of a power rule Use the rules of eponents to simplif each epression. Write the answer with positive eponents onl. Assume all variables represent nonzero real numbers. a) ( ) 5 b) ( ) c) ( ) 5 d) a) ( ) 5 5 Power of a power rule b) ( ) Power of a power rule Definition of a negative eponent ( ) ( ) c) ( ) 5 5 Power of a power rule Product rule d) ( ) 9 ( ) Power of a power rule 7 Quotient rule
2 5. The Power Rules (5) 9 You can use a graphing to illustrate the power of a product rule. Raising a Product to a Power Consider how we would simplif a product raised to a positive power and a product raised to a negative power using known rules. factors of () 8 (a) a (a) (a)(a)(a) a In each of these cases the original eponent is applied to each factor of the product. These eamples illustrate the power of a product rule. Power of a Product Rule If a and b are nonzero real numbers and n is an integer, then (ab) n a n b n. E X A M P L E You can use a graphing to illustrate the power of a quotient rule. Using the power of a product rule Simplif. Assume the variables represent nonzero real numbers. Write the answers with positive eponents onl. a) () 4 b) ( ) c) ( ) a) () 4 () 4 4 Power of a product rule 8 4 b) ( ) () ( ) Power of a product rule 8 Power of a power rule c) ( ) () ( ) ( ) Raising a Quotient to a Power Now consider an eample of appling known rules to a power of a quotient: We get a similar result with a negative power: In each of these cases the original eponent applies to both the numerator and denominator. These eamples illustrate the power of a quotient rule. 5 Power of a Quotient Rule If a and b are nonzero real numbers and n is an integer, then a n a n. b n b
3 70 (54) Chapter 5 Eponents and Polnomials E X A M P L E Using the power of a quotient rule Use the rules of eponents to simplif each epression. Write our answers with positive eponents onl. Assume the variables are nonzero real numbers. a) b) c) d) 4 helpful hint The eponent rules in this section appl to epressions that involve onl multiplication and division. This is not too surprising since eponents, multiplication, and division are closel related. Recall that a a a a and a b a b. a) Power of a quotient rule 8 () b) 9 Because ( ) 9 and ( ) c) 8 d) 4 ) 4 (4 ( ) 9 A fraction to a negative power can be simplified b using the power of a quotient rule as in Eample. Another method is to find the reciprocal of the fraction first, then use the power of a quotient rule as shown in the net eample. E X A M P L E 4 Negative powers of fractions Simplif. Assume the variables are nonzero real numbers and write the answers with positive eponents onl. a) 4 b) 5 c) a) 4 4 The reciprocal of 4 is 4. b) 5 4 Power of a quotient rule ( 5 ) 4 c) 9 4 Variable Eponents So far, we have used the rules of eponents onl on epressions with integral eponents. However, we can use the rules to simplif epressions having variable eponents that represent integers. E X A M P L E 5 Epressions with variables as eponents Simplif. Assume the variables represent integers. a) 4 5 b) (5 ) 5n c) n m
4 5. The Power Rules (55) 7 Did we forget to include the rule (a b) n a n b n?you can easil check with a that this rule is not correct. a) Product rule: b) (5 ) 5 Power of a power rule: n c) 5n ( m n ) 5n Power of a quotient rule ( m ) 5n 5n Power of a power rule 5mn Summar of the Rules The definitions and rules that were introduced in the last two sections are summarized in the following bo. For these rules m and n are integers and a and b are nonzero real numbers.. a n Definition of negative eponent a n. a n a n, a a, and an Negative eponent rules. a 0 Definition of zero eponent 4. a m a n a mn Product rule m 5. a an a mn Quotient rule. (a m ) n a mn Power of a power rule 7. (ab) n a n b n Power of a product rule 8. a b n a b n n Rules for Integral Eponents a n Power of a quotient rule helpful hint In this section we use the amount formula for interest compounded annuall onl. But ou probabl have mone in a bank where interest is compounded dail. In this case r represents the dail rate (APR5) and n is the number of das that the mone is on deposit. E X A M P L E Applications Both positive and negative eponents occur in formulas used in investment situations. The amount of mone invested is the principal, and the value of the principal after a certain time period is the amount. Interest rates are annual percentage rates. Amount Formula The amount A of an investment of P dollars with interest rate r compounded annuall for n ears is given b the formula A P( r) n. Finding the amount According to Fidelit Investments of Boston, U.S. common stocks have returned an average of 0% annuall since 9. If our greatgrandfather had invested $00 in the stock market in 9 and obtained the average increase each ear, then how much would the investment be worth in the ear 00 after 80 ears of growth?
5 7 (5) Chapter 5 Eponents and Polnomials With a graphing ou can enter 00( 0.0) 80 almost as it appears in print. Use n 80, P $00, and r 0.0 in the amount formula: A P( r) n A 00( 0.0) 80 00(.) 80 04,840.0 So $00 invested in 9 would have amounted to $04,840.0 in 00. When we are interested in the principal that must be invested toda to grow to a certain amount, the principal is called the present value of the investment. We can find a formula for present value b solving the amount formula for P: A P( r) n A P Divide each side b ( r) n. ( r) n P A( r) n Present Value Formula Definition of a negative eponent The present value P that will amount to A dollars after n ears with interest compounded annuall at annual interest rate r is given b P A( r) n. E X A M P L E 7 Finding the present value If our greatgrandfather wanted ou to have $,000,000 in 00, then how much could he have invested in the stock market in 9 to achieve this goal? Assume he could get the average annual return of 0% (from Eample ) for 80 ears. Use r 0.0, n 80, and A,000,000 in the present value formula: P A( r) n P,000,000( 0.0) 80 P,000,000(.) 80 Use a with an eponent ke. P A deposit of $488.9 in 9 would have grown to $,000,000 in 80 ears at a rate of 0% compounded annuall. WARMUPS True or false? Eplain our answer. Assume all variables represent nonzero real numbers.. ( ) 5 False. ( ) 8 True. ( ) 9 True 4. ( ) 7 False 5. () False. ( ) 9 9 False 7. True True 7 True 9. 8 True
6 5. The Power Rules (57) 7 5. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the power of a power rule? The power of a power rule sas that (a m ) n a mn.. What is the power of a product rule? The power of a product rule sas that (ab) m a m b m.. What is the power of a quotient rule? The power of a quotient rule sas that (ab) m a m b m. 4. What is principal? Principal is the amount of mone invested initiall. 5. What formula is used for computing the amount of an investment for which interest is compounded annuall? To compute the amount A when interest is compounded annuall, use A P( i) n, where P is the principal, i is the annual interest rate, and n is the number of ears.. What formula is used for computing the present value of an amount in the future with interest compounded annuall? To compute the present value P for the amount A in n ears at annual interest rate i, use P A( i) n. For all eercises in this section, assume the variables represent nonzero real numbers and use positive eponents onl in our answers. Use the rules of eponents to simplif each epression. See Eample. 7. ( ) 8. ( ) 9. ( ) ( ). ( ) 4. ( ) (m ) 4. (a ) 5. ( ) ( ) m 8 a 9. (m ) (m ) 4 7. ( 4 ) 8. ( a ) 5 ( ) ( a ) 4 5 m a Simplif. See Eample. 9. (9) 8 0. (a) 8a. (5w ) 5w. (w 5 ) w ( ) 4. (a b ) a b 5. (ab ) b. ( ) 9 a 8 7. ab 8. 5a b ( ) (5 ab ) 9. ( ab) 0. ( ) ab 8a b Simplif. See Eample.. w w. 8 m 5 m 5. a 4 7a b 8b a b 7 8a b Simplif. See Eample a b c 4. a b a b c a b a b 8 Simplif each epression. Assume that the variables represent integers. See Eample t 5 4t 5 t 48. n 4n 49. ( w ) w w ( ) m p 5. 7 m 7 7 m p 4p 5. 8 a (8 a4 ) 8 5a 54. (5 4 ) (5 ) 5 87 Use the rules of eponents to simplif each epression. If possible, write down onl the answer ( 4 ) 57. ( ) z z ( ). ( ) 7 Use the rules of eponents to simplif each epression ( 4 5a b ) b 70. (m n ) 4 4 ( 5ab ) a mn 7 m n 7
7 74 (58) Chapter 5 Eponents and Polnomials 7. ( ) ( ) (7) 7. ( ) ( (9 9 5 ) ) b 7. a 4 c (a b ) 74. (7z ) 4 7 z ac 8 7 z Write each epression as raised to a power. Assume that the variables represent integers n 80. n5 5n n n5 8. m n m 8. n 8 4m n Use a to evaluate each epression. Round approimate answers to three decimal places (. 5) (. 5) (0.0) (4.9) 88. (4.7) 5(0.47) (5.7) ( 4. 9) 90. [5.9 (0.74) ] (. 7) Solve each problem. See Eamples and Deeper in debt. Melissa borrowed $40,000 at % compounded annuall and made no paments for ears. How much did she owe the bank at the end of the ears? (Use the compound interest formula.) $5, Comparing stocks and bonds. According to Fidelit Investments of Boston, throughout the 990s annual returns on common stocks averaged 9%, whereas annual returns on bonds averaged 9%. a) If ou had invested $0,000 in bonds in 990 and achieved the average return, then what would our investment be worth after 0 ears in 000? $,7.4 Value of $0,000 investment (in thousands of dollars) Stocks Bonds Number of ears after 990 FIGURE FOR EXERCISE 9 b) How much more would our $0,000 investment be worth in 000 if ou had invested in stocks? $, Saving for college. Mr. Watkins wants to have $0,000 in a savings account when his little Wanda is read for college. How much must he deposit toda in an account paing 7% compounded annuall to have $0,000 in 8 ears? $, Saving for retirement. In the 990s returns on Treasur Bills fell to an average of 4.5% per ear (Fidelit Investments). Wilma wants to have $,000,000 when she retires in 45 ears. If she assumes an average annual return of 4.5%, then how much must she invest now in Treasur Bills to achieve her goal? $75, Life epectanc of white males. Strange as it ma seem, our life epectanc increases as ou get older. The function L 7.(.00) a can be used to model life epectanc L for U.S. white males with present age a (National Center for Health Statistics, a) To what age can a 0earold white male epect to live? 75. ears b) To what age can a 0earold white male epect to live? (See also Chapter Review Eercises 5 and 54.) 8.4 ears 9. Life epectanc of white females. Life epectanc improved more for females than for males during the 940s and 950s due to a dramatic decrease in maternal mortalit rates. The function L 78.5(.00) a can be used to model life epectanc L for U.S. white females with present age a. a) To what age can a 0earold white female epect to live? 80. ears b) Bob, 0, and Ashle,, are an average white couple. How man ears can Ashle epect to live as a widow? 7.9 ears c) Wh do the life epectanc curves intersect in the accompaning figure? At 80 both males and females can epect about 5 more ears. Life epectanc (ears) White females White males Present age FIGURE FOR EXERCISES 95 AND 9
8 5. Addition, Subtraction, and Multiplication of Polnomials (59) 75 GETTING MORE INVOLVED 97. Discussion. For which values of a and b is it true that (ab) a b? Find a pair of nonzero values for a and b for which (a b) a b. 98. Writing. Eplain how to evaluate in three different was. 99. Discussion. Which of the following epressions has a value different from the others? Eplain. a) b) 0 c) d) () e) () d 00. True or False? Eplain our answer. a) The square of a product is the product of the squares. b) The square of a sum is the sum of the squares. a) True b) False GRAPHING CALCULATOR EXERCISES 0. At % compounded annuall the value of an investment of $0,000 after ears is given b 0,000(.). a) Graph 0,000(.) and the function 0,000 on a graphing. Use a viewing window that shows the intersection of the two graphs. b) Use the intersect feature of our to find the point of intersection. c) The coordinate of the point of intersection is the number of ears that it will take for the $0,000 investment to double. What is that number of ears? b) (., 0,000) c). ears 0. The function 7.(.00) gives the life epectanc of a U.S. white male with present age. (See Eercise 95.) a) Graph 7.(.00) and 8 on a graphing. Use a viewing window that shows the intersection of the two graphs. b) Use the intersect feature of our to find the point of intersection. c) What does the coordinate of the point of intersection tell ou? b) (87.54, 8) c) At ears of age ou can epect to live until 8. The model fails here. In this section Polnomials Evaluating Polnomials Addition and Subtraction of Polnomials Multiplication of Polnomials 5. ADDITION, SUBTRACTION, AND MULTIPLICATION OF POLYNOMIALS A polnomial is a particular tpe of algebraic epression that serves as a fundamental building block in algebra. We used polnomials in Chapters and, but we did not identif them as polnomials. In this section ou will learn to recognize polnomials and to add, subtract, and multipl them. Polnomials The epression 5 7 is an eample of a polnomial in one variable. Because this epression could be written as (5 ) 7 (), we sa that this polnomial is a sum of four terms:, 5, 7, and. A term of a polnomial is a single number or the product of a number and one or more variables raised to whole number powers. The number preceding the variable in each term is called the coefficient of that variable. In 5 7 the coefficient of is, the coefficient of is 5, and the coefficient of is 7. In algebra a number is frequentl referred to as a constant, and so the last term is called the constant term. A polnomial is defined as a single term or a sum of a finite number of terms.
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