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478 (9–12) Chapter 9 In this Radicals and Rational Exponents 9.2 RATIONAL EXPONENTS section ● Rational Exponents ● Using the Rules of Exponents ● Simplifying Expressions Involving Variables You have learned how to use exponents to express powers of numbers and radicals to express roots. In this section you will see that roots can be expressed with exponents also. The advantage of using exponents to express roots is that the rules of exponents can be applied to the expressions. Rational Exponents The nth root of a number can be expressed by using radical notation or the exponent 3 8 both represent the cube root of 8, and we have 1n. For example, 813 and calculator 3 8 2. 813 close-up You can find the fifth root of 2 using radical notation or exponent notation. Note that the fractional exponent 15 must be in parentheses. Definition of a1/ n If n is any positive integer, then n a, a1n n a is a real number. provided that Later in this section we will see that using exponent 1n for nth root is compatible with the rules for integral exponents that we already know. E X A M P L E 1 Radicals or exponents Write each radical expression using exponent notation and each exponential expression using radical notation. 3 35 a) 4 b) xy Solution 3 a) 3513 35 b) xy (xy)14 4 c) 512 d) a15 c) 512 5 d) a15 a ■ 5 In the next example we evaluate some exponential expressions. E X A M P L E 2 Finding roots Evaluate each expression. b) (8)13 a) 412 c) 8114 d) (9)12 Solution a) 412 4 2 3 13 2 b) (8) 8 4 3 c) 8114 81 is an even root of a negative number, it is not a real d) Because (9)12 or 9 ■ number. 9.2 Rational Exponents (9–13) 479 We now extend the definition of exponent 1n to include any rational number as an exponent. The numerator of the rational number indicates the power, and the denominator indicates the root. For example, the expression ↓| 823 ← Power Root represents the square of the cube root of 8. So we have 823 (813)2 (2)2 4. helpful hint Note that in amn we do not require mn to be reduced. As long as the nth root of a is real, then the value of amn is the same whether or not mn is in lowest terms. Definition of am/ n If m and n are positive integers, then amn (a1n)m , provided that a1n is a real number. We define negative rational exponents just like negative integral exponents. Definition of am/ n If m and n are positive integers and a 0, then 1 amn m, a n provided that a1n is a real number. E X A M P L E 3 Radicals or exponents Write each radical expression using exponent notation and each exponential expression using radical notation. 1 3 x2 b) 3 a) 4 m c) 523 d) a25 Solution 1 1 b) 3 34 m34 4 m m 1 d) a25 2 5 a 3 a) x2 x 23 3 c) 523 52 ■ To evaluate an expression with a negative rational exponent, remember that the denominator indicates root, the numerator indicates power, and the negative sign indicates reciprocal: amn ↑↑ ↑ Root Power Reciprocal 480 (9–14) Chapter 9 Radicals and Rational Exponents The root, power, and reciprocal can be evaluated in any order. However, to evaluate amn mentally it is usually simplest to use the following strategy. Strategy for Evaluating am/ n Mentally 1. Find the nth root of a. 2. Raise your result to the mth power. 3. Find the reciprocal. For example, to evaluate 823 mentally, we find the cube root of 8 (which is 2), 1 square 2 to get 4, then find the reciprocal of 4 to get . In print 823 could be 4 1 written for evaluation as ((813)2)1 or 132. (8 ) E X A M P L E 4 calculator close-up A negative fractional exponent indicates a reciprocal, a root, and a power. To find 432 you can find the reciprocal first, the square root first, or the third power first as shown here. Rational exponents Evaluate each expression. b) 432 a) 2723 c) 8134 d) (8)53 Solution a) Because the exponent is 23, we find the cube root of 27 and then square it: 2723 (2713)2 32 9 b) Because the exponent is 32, we find the square root of 4, cube it, and find the reciprocal: 1 1 1 432 12 (4 )3 23 8 c) Because the exponent is 34, we find the fourth root of 81, cube it, and find the reciprocal: 1 1 1 8134 14 3 3 3 27 (81 ) Definition of negative exponent 1 1 1 1 d) (8)53 13 5 5 ((8) ) (2) 32 32 ■ CAUTION An expression with a negative base and a negative exponent can have a positive or a negative value. For example, 1 (8)53 32 and 1 (8)23 . 4 Using the Rules of Exponents All of the rules for exponents hold for rational exponents as well as integral exponents. Of course, we cannot apply the rules of exponents to expressions that are not real numbers. 9.2 Rational Exponents (9–15) 481 Rules for Rational Exponents The following rules hold for any nonzero real numbers a and b and rational numbers r and s for which the expressions represent real numbers. 1. aras ar s ar 2. s ars a 3. (ar ) s ars 4. (ab)r arbr a r ar 5. r b b Product rule Quotient rule Power of a power rule Power of a product rule Power of a quotient rule We can use the product rule to add rational exponents. For example, 1614 1614 1624. The fourth root of 16 is 2, and 2 squared is 4. So 1624 4. Because we also have 1612 4, we see that a rational exponent can be reduced to its lowest terms. If an exponent can be reduced, it is usually simpler to reduce the exponent before we evaluate the expression. We can simplify 1614 1614 as follows: 1614 1614 1624 1612 4 E X A M P L E 5 Using the product and quotient rules with rational exponents Simplify each expression. 534 b) 1 a) 2716 2712 5 4 Solution a) 2716 2712 2716 12 2723 9 3 4 5 b) 1 53414 524 512 5 5 4 E X A M P L E 6 Product rule for exponents We used the quotient rule to subtract the exponents. ■ Using the power rules with rational exponents Simplify each expression. a) 312 1212 b) (310 )12 26 c) 9 3 13 Solution a) Because the bases 3 and 12 are different, we cannot use the product rule to add the exponents. Instead, we use the power of a product rule to place the 12 power outside the parentheses: 312 1212 (3 12)12 3612 6 482 (9–16) Chapter 9 Radicals and Rational Exponents b) Use the power of a power rule to multiply the exponents: (310)12 35 26 c) 9 3 13 (26)13 (39)13 22 3 3 33 2 2 27 4 Power of a quotient rule Power of a power rule Definition of negative exponent ■ Simplifying Expressions Involving Variables helpful hint We usually think of squaring and taking a square root as inverse operations, which they are as long as we stick to positive numbers.We can square 3 to get 9, and then find the square root of 9 to get 3— what we started with. We don’t get back to where we began if we start with 3. When simplifying expressions involving rational exponents and variables, we must be careful to write equivalent expressions. For example, in the equation (x2)12 x it looks as if we are correctly applying the power of a power rule. However, this statement is false if x is negative because the 12 power on the left-hand side indicates the positive square root of x2. For example, if x 3, we get [(3)2]12 912 3, which is not equal to 3. To write a simpler equivalent expression for (x 2)12, we use absolute value as follows. Square Root of x 2 (x 2)12 x for any real number x. x 2 x . Both of these equations are Note that (x 2)12 x is also written as identities. It is also necessary to use absolute value when writing identities for other even roots of expressions involving variables. E X A M P L E 7 Using absolute value symbols with roots Simplify each expression. Assume the variables represent any real numbers and use absolute value symbols as necessary. x9 13 b) a) (x8y4)14 8 Solution a) Apply the power of a product rule to get the equation (x8y4)14 x 2y. The lefthand side is nonnegative for any choices of x and y, but the right-hand side is negative when y is negative. So for any real values of x and y we have (x8y4)14 x2 y . 9.2 Rational Exponents (9–17) 483 b) Using the power of a quotient rule, we get x9 8 13 x3 . 2 This equation is valid for every real number x, so no absolute value signs are ■ used. Because there are no real even roots of negative numbers, the expressions a12, x34, and y16 are not real numbers if the variables have negative values. To simplify matters, we sometimes assume the variables represent only positive numbers when we are working with expressions involving variables with rational exponents. That way we do not have to be concerned with undefined expressions and absolute value. E X A M P L E 8 Expressions involving variables with rational exponents Use the rules of exponents to simplify the following. Write your answers with positive exponents. Assume all variables represent positive real numbers. a12 b) 1 a) x23x43 a 4 x2 12 c) (x12y3)12 d) 1 y 3 Solution a) x23x43 x63 Use the product rule to add the exponents. x2 Reduce the exponent. 12 a b) 1 a1214 Use the quotient rule to subtract the exponents. a 4 a14 Simplify. 12 3 12 12 12 3 12 c) (x y ) (x ) (y ) Power of a product rule 14 32 x y Power of a power rule 14 x 3 Definition of negative exponent y 2 d) Because this expression is a negative power of a quotient, we can first find the reciprocal of the quotient, then apply the power of a power rule: x2 1 y 3 12 y13 x2 12 y16 x 1 1 1 3 2 6 WARM-UPS True or false? Explain your answer. 3 1. 913 9 5 2. 853 83 3. (16)12 1612 6 5. 612 6 7. 212 212 412 4. 932 27 2 6. 12 212 2 8. 1614 2 9. 616 616 613 1 10. (28)34 26 ■ 484 (9–18) 9.2 Chapter 9 Radicals and Rational Exponents EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. How do we indicate an nth root using exponents? 37. 1632 38. 2532 39. (27)13 40. (8)43 2. How do we indicate the mth power of the nth root using exponents? 41. (16)14 42. (100)32 3. What is the meaning of a negative rational exponent? Use the rules of exponents to simplify each expression. See Examples 5 and 6. 43. 313314 44. 212 213 13 13 45. 3 3 46. 514514 13 8 2723 47. 2 48. 8 3 27 13 4. Which rules of exponents hold for rational exponents? 5. In what order must you perform the operations indicated by a negative rational exponent? 6. When is amn a real number? Write each radical expression using exponent notation and each exponential expression using radical notation. See Example 1. 4 7. 7 3 8. cbs 9. 915 11. 5x 13. a12 10. 312 12. 3y 14. (b)15 Evaluate each expression. See Example 2. 15. 17. 19. 21. 22. 2512 (125)13 1614 (4)12 (16)14 16. 1612 18. (32)15 20. 813 Write each radical expression using exponent notation and each exponential expression using radical notation. See Example 3. 3 w7 23. 1 25. 3 10 2 34 24. a5 26. a1 3 53 28. 6 29. (ab)32 30. (3m)15 Evaluate each expression. See Example 4. 33. 25 32 35. 2743 50. 914 934 51. 1812 212 53. (26)13 52. 812212 54. (310)15 55. (38)12 56. (36)13 57. (24)12 58. (54)12 34 59. 6 2 54 60. 6 3 12 12 Simplify each expression. Assume the variables represent any real numbers and use absolute value as necessary. See Example 7. 61. (x 4)14 62. (y6)16 8 12 63. (a ) 64. (b10)12 65. (y3)13 66. (w9)13 67. (9x6y2)12 81x12 14 69. 2 y0 68. (16a 8b 4)14 144a8 12 70. 9y18 Simplify. Assume all variables represent positive numbers. Write answers with positive exponents only. See Example 8. 71. x12x14 72. y13y13 73. (x12 y)(x34y12) 74. (a12b13)(ab) w 13 75. w3 77. (144x 16 )12 a12 4 79. b14 a12 76. a2 78. (125a 8)13 2a12 6 80. 1 b 3 2 27. w 31. 12523 49. 434 414 32. 100023 32 34. 16 36. 1634 Simplify each expression. Write your answers with positive exponents. Assume that all variables represent positive real numbers. 81. (92)12 82. (416 )12 9.2 83. 1634 84. 2532 85. 12543 86. 2723 87. 212214 88. 91912 89. 30.26 30.74 91. 3142714 8 23 93. 27 1 34 95. 16 90. 21.520.5 92. 323923 8 13 94. 27 9 12 96. 16 (9–19) Rational Exponents 485 In Exercises 123–130, solve each problem. 123. Diagonal of a box. The length of the diagonal of a box can be found from the formula D (L2 W2 H 2)12, where L, W, and H represent the length, width, and height of the box, respectively. If the box is 12 inches long, 4 inches wide, and 3 inches high, then what is the length of the diagonal? D 9 12 9 13 97. (9x ) 98. (27x ) 99. (3a23)3 100. (5x12)2 4 in. 3 in. 12 in. FIGURE FOR EXERCISE 123 101. (a12b)12(ab12) 102. (m14n12)2(m2n3)12 103. (km12)3(k 3m5)12 104. (tv13)2(t 2 v3)12 Use a scientific calculator with a power key (xy ) to find the decimal value of each expression. Round answers to four decimal places. 105. 213 106. 512 107. 2 108. (3) 109. 1024110 110. 77760.2 111. 80.33 112. 2890.5 12 113. 64 15,625 0.75V r 13 . Find the radius of a spherical tank that has a volume of 32 cubic meters. 3 13 16 32 114. 243 r 35 Simplify each expression. Assume a and b are positive real numbers and m and n are rational numbers. 116. bn2 bn3 115. am2 am4 am5 117. am3 bn4 118. bn3 119. (a1mb1n)mn 120. (am2bn3)6 a3mb6n 13 124. Radius of a sphere. The radius of a sphere is a function of its volume, given by the formula 121. 9 am a3mb6n 13 122. a6mb9n FIGURE FOR EXERCISE 124 125. Maximum sail area. According to the new International America’s Cup Class Rules, the maximum sail area in square meters for a yacht in the America’s Cup race is given by S (13.0368 7.84D13 0.8L)2, where D is the displacement in cubic meters (m3), and L is the length in meters (m). (Scientific American, May 1992). Find the maximum sail area for a boat that has a displacement of 18.42 m3 and a length of 21.45 m. 486 (9–20) Chapter 9 Radicals and Rational Exponents S (m2) 128. Best bond fund. The top bond fund for 1997 in the 5-year category was GT Global High Income B. An investment of $10,000 in 1992 grew to $21,830.95 in 1997. Use the formula from the previous exercise to find the 5-year average annual return for this fund. D (m3) L (m) FIGURE FOR EXERCISE 125 126. Orbits of the planets. According to Kepler’s third law of planetary motion, the average radius R of the orbit of a planet around the sun is determined by R T 23, where T is the number of years for one orbit and R is measured in astronomical units or AUs (Windows to the Universe, www.windows.umich.edu). a) It takes Mars 1.881 years to make one orbit of the sun. What is the average radius (in AUs) of the orbit of Mars? b) The average radius of the orbit of Saturn is 9.05 AU. Use the accompanying graph to estimate the number of years it takes Saturn to make one orbit of the sun. 129. Overdue loan payment. In 1777 a wealthy Pennsylvania merchant, Jacob DeHaven, lent $450,000 to the Continental Congress to rescue the troops at Valley Forge. The loan was not repaid. In 1990 DeHaven’s descendants filed suit for $141.6 billion (New York Times, May 27, 1990). What average annual rate of return were they using to calculate the value of the debt after 213 years? (See Exercise 127.) 130. California growin’. The population of California grew from 19.9 million in 1970 to 32.5 million in 2000 (U.S. Census Bureau, www.census.gov). Find the average annual rate of growth for that time period. (Use the formula from Exercise 127 with P being the initial population and S being the population n years later.) California population (millions of people) 35 Radius of orbit (AU) 10 8 R T 2/3 6 4 30 25 20 15 10 2 0 10 20 Time for one orbit (years) 30 FIGURE FOR EXERCISE 130 FIGURE FOR EXERCISE 126 127. Best stock fund. The average annual return for an investment is given by the formula S r P 1970 1980 1990 2000 Year 1n 1, where P is the initial investment and S is the amount it is worth after n years. The top mutual fund for 1997 in the 3-year category was Fidelity Select-Energy Services (Money Guide to Mutual Funds, 1998), in which an investment of $10,000 grew to $31,895.06 from 1994 to 1997. Find the 3-year average annual return for this fund. GET TING MORE INVOLVED 131. Discussion. If we use the product rule to simplify (1)12 (1)12, we get (1)12 (1)12 (1)1 1. If we use the power of a product rule, we get (1)12 (1)12 (1 1)12 112 1. Which of these computations is incorrect? Explain your answer. 132. Discussion. Determine whether each equation is an identity. Explain. a) (w 2 x 2)12 w x b) (w 2x 2)12 wx c) (w 2x 2)12 w x