LESSON 6.1 EXPONENTS
Overview
Rosa plans to invest $1000 in an Individual Retirement Account (IRA). She can invest in bonds that offer a return of 7% annually, or a riskier stock fund that is expected to return 10% annually. Rosa would like to know how much her money can grow in 30 years. Exponents can help her answer this question. In this lesson, you will study exponents and their properties.
Explain
Concept 1 has sections on CONCEPT 1: • Exponential Notation PROPERTIES OF EXPONENTS •Multiplication Property
• Division Property Exponential Notation Exponents are used to indicate repeated multiplication of the same • Power of a Power Property number. • Power of a Product For example, we use exponential notation to write: Property 5 5 5 5 54 • Power of a Quotient Property 54 is read “five to the fourth power.”
• Zero Power Property In the expression 54:
•Using Several Properties of • The base, 5, is the repeated factor. Exponents • The exponent, 4, indicates the number of times the base appears as a factor. An exponent is also called a power.
Exponent Base 54 5 5 5 5 625 4 factors Product
LESSON 6.1 EXPONENTS EXPLAIN 367 Example 6.1.1 Find: 23 Solution The base is 2 and the exponent is 3. 23 = 2 2 2 = 8 3 factors
Example 6.1.2
Rewrite using exponential notation: 10 10 10 10 10 10 Solution There are six factors. Each is 10. 10 10 10 10 10 10 = 106 Therefore, the base is 10 and the 6 factors exponent is 6.
Exponents have several properties. We will use these properties to simplify expressions. In the properties that follow, each variable represents a real number.
Multiplication Property
— Property — Multiplication Property of Exponents English To multiply two exponential expressions with the same base, add their exponents. The base stays the same. Algebra xm xn xm n (Here, m and n are positive integers.) Example 54 52 54 2 5 6
Example 6.1.3 a. Use the Multiplication Property of Exponents to simplify 23 24. b. Use the definition of exponential notation to justify your answer. Solution Remember to add the exponents, a. The operation is multiplication and the bases are the same. but leave the bases alone. Therefore, add the exponents and use 2 as the base. That is, 23 24 23 4 27, not 47. 23 24 23 4 27 Note the difference between b. Rewrite the product to show the factors. Then simplify. 23 24 and 23 24. 23 24 = (2 2 2) (2 2 2 2) = 2 2 2 2 2 2 2 = 27 23 24 23 4 27 128 23 24 8 16 24 3 factors 4 factors 7 factors
368 TOPIC 6 EXPONENTS AND POLYNOMIALS — Caution — Negative Bases A negative sign is part of the base only when the negative sign is inside the parentheses that enclose the base. For example, consider the following cases: In ( 3)2, the base is 3. In 32, the base is 3. ( 3)2 ( 3) ( 3) 9You can think of 32 as the “opposite” of 32. 32 (3 3) 9
Example 6.1.4 If possible, use the Multiplication Property of Exponents to simplify each expression: a. ( 2)2 ( 2)4 b. 22 24 c. 22 ( 2)4
Solution a. In ( 2)2 ( 2)4, the base ( 2)2 ( 2)4 ( 2)2 4 is 2. ( 2)6 64 b. In 22 24, the base is 2. 22 24 (22) (24) We may think of 22 24 (22 4) as the opposite of 22 24. (26) 64 c. In 22 ( 2)4, the base of the first factor, 22, is 2. The base of the second factor, ( 2)4, is 2. The bases are not the same, so we cannot use the Multiplication Property of Exponents. However, we can still evaluate the expression. 22 ( 2)4 4 16 64
We can extend the Multiplication Property of Exponents to multiply more than two factors.
Example 6.1.5 Find: 84 8 85. Leave your answer in exponential notation. We left 810 in exponential form. To Solution evaluate 810, use the “yx ” key on a The bases are the same, so we can use the scientific calculator or the “^” key Multiplication Property of Exponents. on a graphing calculator. 10 Note: 8 81 84 8 85 84 81 85 84 1 5 810 8 1,073,741,824
LESSON 6.1 EXPONENTS EXPLAIN 369 Example 6.1.6 Find: x7 x3 x5 Solution The operation is multiplication and the bases are the same. Therefore, add the exponents and x7 x3 x5 x7 3 5 x15 use x as the base.
Division Property
— Property — Division Property of Exponents English To divide two exponential expressions with the same base: Compare the exponents. • If the greater exponent is in the numerator, write the base in the numerator. • If the greater base is in the denominator, write the base in the denominator. Then subtract the smaller exponent from the greater. Use the result as the new exponent. Algebra Example xm 25 xm n for m n and x 0. 25 3 22 xn 23 xm 1 23 1 1 for m n and x 0. xn xn m 25 25 3 22 (Here, m and n are positive integers.)
Example 6.1.7
xm 1 53 Since 3 4, we use the form . a. Use the Division Property of Exponents to find . xn xn m 54 b. Use the definition of exponential notation to justify your answer. Solution 53 1 1 a. The bases are the same, ¬ 54 54 3 5 so subtract the exponents. 53 5 5 5 b. Rewrite the numerator and ¬ 54 5 5 5 5 denominator to show the factors. 1 1 1 5 5 5 1 Cancel the common factors. ¬ 5 5 5 5 5 1 1 1
370 TOPIC 6 EXPONENTS AND POLYNOMIALS Example 6.1.8 Find: 79 76. Leave your answer in exponential notation.
Solution The operation is division and the 9 xm 9 6 7 9 6 3 m n bases are the same. Therefore, subtract 7 7 7 7 Since 9 6, we use the form n x . 76 x the exponents and use 7 as the base.
Example 6.1.9 Find: w8 w13
Solution
The operation is division and the 8 m 8 13 w 1 1 x 1 bases are the same. Therefore, subtract w w Since 8 13, we use the form n n m . w13 w13 8 w5 x x the exponents and use w as the base.
Power of a Power Property
— Property — Power of a Power Property of Exponents English To raise a power to a power, multiply the exponents. Algebra (xm)n xmn (Here, m and n are positive integers.) Example (72)4 72 4 78
Example 6.1.10 a. Use the Power of a Power Property of Exponents to simplify (52)3. b. Use the definition of exponential notation to justify your answer. Solution a. To raise a power to a power, (52)3 52 3 56 multiply the exponents. b. Rewrite each power to show the factors. Then simplify. 3 ( 52) = ( 52) ( 52) ( 52) = (5 5) (5 5) (5 5) = 56 3 factors 6 factors
LESSON 6.1 EXPONENTS EXPLAIN 371 Example 6.1.11 Simplify: (y5)3 Solution To simplify a power of a power, (y5)3 y5 3 y15 multiply the exponents.
Power of a Product Property
— Property — Power of a Product Property of Exponents English To raise a product to a power, you can first raise each factor to the power. Then multiply. Algebra (xy)n xnyn (Here, n is a positive integer.) Example (2x)3 23x3 8x3
Example 6.1.12 a. Use the Power of a Product Property of Exponents to simplify (3y)2. b. Use the definition of exponential notation to justify your answer. Solution a. Raise each factor to the power 2. (3y)2¬ 32y2 9y2 b. Rewrite the power to show (3y)2¬ (3y) (3y) the factors. Then simplify. ¬ 3 3 y y ¬ 32y2 ¬ 9y2
Example 6.1.13 Simplify: (23 w5)4 Solution Use the Power of a Product Property of Exponents to raise each factor inside (23 w5)4¬ (23)4(w5)4 the parentheses to the power 4. Use the Power of a Power Property ¬ (23 4)(w5 4) of Exponents.
We left 212 in exponential form. To Simplify. ¬ 212w20 evaluate 212, use the “yx ” key on a scientific calculator or the “^” key on a graphing calculator.
2 12 4096
372 TOPIC 6 EXPONENTS AND POLYNOMIALS Power of a Quotient Property
— Property — Power of a Quotient Property of Exponents English To raise a quotient to a power, you can first raise the numerator and denominator each to the power. Then divide. x n xn Algebra , y 0 y yn (Here, n is a positive integer.) 4 4 Example 2 2 1 6 x x4 x4
Example 6.1.14 2 3 a. Use the Power of a Quotient Property of Exponents to simplify . 5 b. Use the definition of exponential notation to justify your answer. Solution 2 3 23 8 a. Raise the numerator to the power 3. 5 53 125 Raise the denominator to the power 3. b. Rewrite the power to show the factors. Then simplify. 3 3 2 2 2 2 2 2 2 2 8 5 5 5 5 5 5 5 53 125 3 factors
Zero Power Property
— Property — Zero Power Property English Any real number, except zero, raised to the power 0 is 1. Algebra x0 1, x 0 Example 170 1
Here’s a way to understand why 170 is 1. This same reasoning applies no matter what power or nonzero base we choose. Suppose we write 0 as 2 2. xn 0 2 2 xn n x0 Then, 17 17 . xn 1 1 172 1 7 17 xn By the Division Property of Exponents, 172 2 1. 1 172 17 17 xn 1 1 Therefore, x0 1 for x 0. Since 170 172 2 and 172 2 1, we have 170 1.
LESSON 6.1 EXPONENTS EXPLAIN 373 Example 6.1.15 a. Use the Zero Power Property to simplify 50. b. Justify your answer. Solution a. Any real number, except zero, 50¬ 1 raised to the power 0 is 1. 53 b. Suppose we have . 53 53 We can simplify this using the ¬ 53 3 50 53 Division Property of Exponents. 1 1 1 53 53 5 5 5 But if we reduce the fraction , ¬ 1 53 53 5 5 5 the result is 1. 1 1 1 53 Since is equivalent to both 50 and 1, 53 we conclude 50 1.
Example 6.1.16 Find each of the following. (Assume each variable represents a nonzero real number). w0 a. ( 7)0 b. c. (12x4y5)0 d. 2y0 e. 00 4 Solution In each case, we apply the Zero Power Property: any nonzero real number raised to the zero power is 1. a. The base is the real number 7. ( 7)0¬ 1 w0 1 b. The base, w, represents a nonzero real number. ¬ 4 4 c. The base, 12x4y5, represents a nonzero (12x4y5)0¬ 1 real number. d. Only y is raised to the power 0. 2y0 2 1¬ 2 e. In the Zero Power Property, 00 is undefined the base cannot be 0.
Using Several Properties of Exponents To simplify an exponential expression, we may need to use several properties of exponents.
374 TOPIC 6 EXPONENTS AND POLYNOMIALS Example 6.1.17
25 x4 3 Find: 28 Solution 25 x4 3 First, we simplify the expression 28 inside the parentheses.
To combine the powers of 2, 4 3 x xm 1 subtract exponents. Since 5 8, we use the form . 23 xn xn m (Division Property of Exponents) Raise the numerator and the (x4)3 denominator each to the power 3. (23)3 (Power of a Quotient Property of Exponents) x4 3 Multiply exponents: 4 3 12 and 3 3 9. 23 3 (Power of a Power Property of Exponents) 12 x 29
Example 6.1.18 (x3 x5y4)3 Find: (y4)2 Solution (x3 x5y4)3 First, we simplify the expression (y4)2 inside the parentheses in the numerator. To combine the powers of x, (x8y4)3 add their exponents. (y4)2 (Multiplication Property of Exponents) In the numerator, raise each (x8)3(y4)3 factor to the power 3. (y4)2 (Power of a Product Property of Exponents) Multiply exponents: 8 3 24 x24y12 and 4 3 = 12 and 4 2 8. y8 (Power of a Power Property of Exponents). To combine the powers of y, xm subtract their exponents. x 24y4 Since 12 > 8, we use the form x m n. xn (Division Property of Exponents)
LESSON 6.1 EXPONENTS EXPLAIN 375 Example 6.1.19 2x2(3y4)2 Find: 6xy5 Solution 2x2(3y4)2 Raise each factor inside the 6xy5 parentheses to the power 2. 2x2(3)2(y4)2 (Power of a Product Property of Exponents) 6xy5 To simplify (y 4 )2, multiply 2x232y8 exponents: 4 2 8. 6xy5 (Power of a Power Property of Exponents) 18x2y8 Multiply the constants: 2 32 2 9 18 6xy5 Divide 18 by 6. To combine the powers of x, subtract their exponents. 3xy3 To combine the powers of y, subtract their exponents. (Division Property of Exponents)
Real world problems often involve exponents. For example, the following formula may be used to calculate the value of an investment after a certain number of years. A P(1 r)t where A is the value of the investment, P is the original principal invested, r is the annual rate of return, and t is the number of years the money is invested.
Example 6.1.20 Rosa plans to invest $1000 in an Individual Retirement Account (IRA). She can invest in a bond fund that averages a 7% annual return, or in a riskier stock fund that is expected to have a 10% annual return. a. Determine the value of the bond fund after 30 years. b. Determine the projected value of the stock fund after 30 years. c. Compare the returns on the two investments.
376 TOPIC 6 EXPONENTS AND POLYNOMIALS Solution For each investment, the principal, A P(1 r)t P, is $1000. The time, t, is 30 years. a. For the bond fund, the annual rate A 1000(1 0.07)30 of return is 7%. So, r 0.07. In the formula, substitute 1000 for P, 0.07 for r, and 30 for t. Add 1 and 0.07. 1000(1.07)30 On a calculator, use the “yx ” key 1000(7.612255043) or the “^” key to approximate 1.0730.
Multiply and round to the nearest $7,612.26 To get a better estimate, we waited until hundredth (cent). the end of the problem to round the answer. After 30 years, the bond fund will be worth $7,612.26. b. For the stock fund, the projected annual A 1000(1 0.10)30 rate of return is 10%. So r 0.10. In the formula, substitute 1000 for P, 0.10 for r, and 30 for t. Add 1 and 0.10. 1000(1.10)30 On a calculator, use the “yx ” key or 1000(17.44940227) the “^” key to approximate 1.1030. Multiply and round to the nearest $17,449.40 hundredth (cent). After 30 years, the stock fund should be worth $17,449.40. c. The bond fund would grow to almost 8 times its original value. The stock fund would grow to over 17 times its original value. The stock fund, which is riskier than the bond fund, is projected to be worth more than twice as much as the bond fund in 30 years.
LESSON 6.1 EXPONENTS EXPLAIN 377 Here is a summary of this concept from Interactive Mathematics.
378 TOPIC 6 EXPONENTS AND POLYNOMIALS Checklist Lesson 6.1
Here is what you should know after completing this lesson.
Words and Phrases exponential notation exponent base power
Ideas and Procedures ❶ Exponential Notation Given an expression written in exponential Example 6.1.1 notation, identify the base, identify the Find: 23 exponent, and evaluate the expression. See also: Example 6.1.2 ❷ Properties of Exponents Use the following properties of exponents to Example 6.1.18 simplify an expression: (x3 x5y4)3 Multiplication Property of Exponents Find: (y4)2 Division Property of Exponents Power of a Power Property of Exponents See also: Example 6.1.3-6.1.17, 6.1.19, 6.1.20 Power of a Product Property of Exponents Apply 1-28 Power of a Quotient Property of Exponents Zero Power Property
LESSON 6.1 EXPONENTS CHECKLIST 379 Homework
Homework Problems
Circle the homework problems assigned to you by the computer, then complete them below.
7. Find: Explain a. (b 3)2 (b 4)3 y6 Properties of Exponents b. (y 5)2 (y 3)4 y17 Use the appropriate properties of exponents to simplify a4 b6 the expressions in questions 1 through 12. (Keep your c. a11 b3 answers in exponential form where possible.) 8. Find: 1. Find: (xy)4 (3b)6 a. b. a. 32 35 b. 52 55 y9 x7 (3b2)4 c. 72 75 9. As animals grow, they get taller faster than they get stronger. In general, this proportion of increase in 2. Find: x2 height to increase in strength can be written as . 39 35 3 a. b. x 35 39 Simplify this fraction. 9 3 10. An animal is proportionally stronger the smaller it c. 9 3 is. If a person is 200 times as tall as an ant, figure 3. Find: out how much stronger a person is, pound for 2002 a. (73)2 b. (72)3 pound, by simplifying the expression . 2003 4. Find: 11. Find: a. (5 x)3 b. (3 y)2 4xy2z 0 y7 y a. b. c. (a 2 b)4 5x2yz3 y9 y2 b3 b5 4 c. d. 2x 0 5y 0 5. Find: b6 b3 x3 x5 2 a12 a6 4 a. b. 12. Find: x4 a9 a7 (x3 x4)2 5 (4a2)0 3b0 b6 b5 3 23 x5 a. b. c. d. x7 2 b3 b8 25 x2 (3x 3x2)2 3 b8 4 6. Find: c. d. 311 x7 (b2 b7) a. (a 2 a 3)2 (a 2 a 3)2 y4 3y2 b. y8 c. x4 x9 x y5 y11
380 TOPIC 6 EXPONENTS AND POLYNOMIALS Apply
Practice Problems
Here are some additional practice problems for you to try.
Properties of Exponents 14. Find: (y 8)3 5 3 1. Find: 7 7 . Leave your answer in exponential 15. Find: (z 12)4 notation. 16. Find: (x 9)4 2. Find: 63 64. Leave your answer in exponential notation. 17. Find: (3 a)4 3. Find: b 12 b 3 18. Find: (4 b)2 4. Find: c 9 c 4 19. Find: (2 y)3
6 5 a6b5 5. Find: a a 20. Find: a8b2 7 3 6. Find: 5 5 . Leave your answer in exponential m7n4 21. Find: notation. m3n10 3 7 12 7. Find: 910 94. Leave your answer in exponential x y z 22. Find: 8 5 notation. xy z 0 m10 23. Find: 5 8. Find: m4 24. Find: 3480 n20 9. Find: 0 n15 25. Find: x b12 1 0 10. Find: 26. Find: 5 (4z) b5 0 0 1 11. Find: (53)4. Leave your answer in exponential 27. Find: a (xyz) 3 notation. 28. Find: 21 (3x)0 y0 12. Find: (82)5. Leave your answer in exponential notation. 13. Find: (135)6. Leave your answer in exponential notation.
LESSON 6.1 EXPONENTS APPLY 381 Evaluate
Practice Test
Take this practice test to be sure that you are prepared for the final quiz in Evaluate.
1. Rewrite each expression below. Keep your answer 5. Simplify each expression below. in exponential form where possible. a. (b 4 b 2)8 a. 11 11 11 11 b. (35 a 6)2 b. 3 3 y y y y y c. (29 x4 y 6)11 c. 512 58 523 6. Simplify each expression below. d. x 7 y y 19 x 14 y 6 5y10 4 a. e. 78 b 5 b 8 710 b 3x8 3 4 6 2. Rewrite each expression below in simplest form 7a b b. 2 using exponents. 5a 2 2 2 2 2 2 7. Calculate the value of each expression below. a. 2 2 2 a. (4x)0 2y 0 b20 2 3 0 b. b. (5xy 4x ) b14 c. 2x 0 y 0 312 x7 0 0 0 c. 9 16 (4x) 3x 2x 3 x d. 2 2 2 y17 d. y14 y3 y4 8. Rewrite each expression below using a single x3 exponent. 3. Circle the expressions below that simplify to . 5 4 5 7 y a a 6 2 11 5 a. x y y x a a3 3 7 2 4 x y y x 3 7 a a 9 7 b. xy x y a4 a5 x6y4 x4y6 4. Circle the expressions below that simplify to 5y. (31x 8)0 5y ( 5y)0 2 5y y 2 (5y ) 5y 5 5 5 y y y y 5 5 y y
382 TOPIC 6 EXPONENTS AND POLYNOMIALS Lesson 6.1 Exponents Apply - Practice Problems Homework 1. 2xy 5xz ; 9y 2 13yz – 8z 2 1a. 37 b. 57 c. 77 3a. 76 b. 76 3a.binomial b. binomial c. trinomial d. monomial 3 3 5a. x 8 b. a 8 c. 1 d. x 7a. b 18 b. y 11 c. b 4 a7 3e. trinomial 1 1 1 9. 11a. 1 b. c. d. 3 2 x y 3 b4 5. 8 7. 9 9. 7 11. 6 13. 84 15. 6x 11x – 8 Apply - Practice Problems 17. 15m 2n 3 2m 2n 2 – 7mn 19. 15a 3b 2 4a 2b – ab 3 1. 78 3. b 15 5. a 11 7. 96 9. n 5 21. 20xy 2z 3 – 30x 2yz 2 10x 3y 3z 23. 4x 3 7x – 8
11. 512 13. 1330 15. z 48 17. 81a 4 25. y 2 6xy 4y 27. 11a 5b 3 – 4a 4b – 9b 29. 15y 5 m 4 19. 8y 3 21. 23. 1 25. 1 27. 3 9 4 8 2 5 3 3 n 6 31. –45a 33. 28x y 35. –6w x y z 7 7 4 5 4 2 Evaluate - Practice Test 37. –6a b 10a b – 12a b 1a. 114 b. 32y 5 c. 543 d. x 21y 26 e. 718 b14 39. 20a 4b 2 10a 4b 3 – 35a 3b 4 – 15a 2b 3 33 1 2a. 2 3 b. b 6 c. d. 6 3 4 5 4 4 3 4 x 9 y 4 41. 12x y – 28x y 8x y – 4x y 2 3 3 2 5 5 3 6 2 7 2 2 2 5 8a b 3x y z 3n p 3. x y and x y 4. (31x 8)0 5y, 5y , and (5y ) 43. 5a b 45. 47. 49. x 3y 7 x 4y 6 y 5y 3c 2w 2mq 2 3 51. 4a 3a 3 53. 7 4x 3y 55. 2 – x z 5a. b 48 b. 310a12 c. 299 x 44y 66 y x 2y 2 4 40 6 6 24 6a. 5 y b. 7 a b 34x 32 56 Evaluate - Practice Test 7a. –1 b. 1 c. –3 d. 1 1. t 2 – s + 5, m5n 4o 3p 2r, and 5 c15 + 3 c11 – 3 7 14 8a. a 35 b. 1 2. w 5x 4 is a monomial. a 35 2x 2 – 36 is a binomial. Lesson 6.2 Polynomial Operations I 1 x 17 + 2 x 12 – 1 is a trinomial. 3 3 3 Homework 27 is a monomial. 1 1. 3 y 3 + 3y 2 – 5 3a. –4y 5 – 2y 3 + 3y + 2 27x 3 – 2x 2y 3 is a binomial. 4 b. 5, 3, 1, 0 c. 5 5. –4v 7 + v 3 + 6v 2 – 5v + 5 x 2 + 3xy – 2 y 2 is a trinomial. 3 7. –7s 3t 3 + 7st 2 – s 2t + 2st –13t + 9 3. 8w 8 + 7w 5 + 3w 3 – 13w 2 – 2 9. 2x 2y + 10xy2 + 4y 3 + 3 4a. 3x 3y – 8x 2y 2 – 5y 3 + xy + y 2 + 19 3 2 2 3 2 11. 4w 2yz + 3w 3 – 4wyz 2 + 6wyz – 4wy 2z + 3 b. 7x y – 8x y + 3y + 5xy – y + 7 8 3 5 13. x 3y 3z 3 15. –3t 4u 4v 15 17. 10p 3r 4 + 5p 4r 5 5. x y w 5 3 2 2 3 2 3 3 5 5 2 7 19. 3xw 21. 3a d 23. 4xy + 5x y or xy (4 + 5xy) 6. 3n p + 2n p – 35n p y 2b 5c 3 3 3 3 4 6 7. 3x yz 2
2 8. 3t u – 1 t 4 2v 2
LESSON 6.2: ANSWERS 727