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Summary 8 DEFINITION /PROCEDURE EXAMPLE REFERENCE Roots and Radicals Section 8.1 Square Roots Every positive number has two square roots. The positive or principal square root of a number a is denoted 1a 125 5 5 is the principal square root of 25 because 52 25. 149 7 The negative square root is written as 1a because (7)2 49. Higher Roots Cube roots, fourth roots, and so on are denoted by using an index and a radical. The principal nth root of a is written as 3 127 3 3 164 4 4 181 3 Index 242 4 n 1a 2(5)2 5 3 Radical sign Radicand 2(3)3 3 2m2 m Radicals Containing Variables In general, x if n is even n 2 xn x if n is odd 3 227x3 3x p. 581 Simplification of Radical Expressions Section 8.2 Simplifying radical expressions entails applying two properties for radicals. Product Property n n n 1 ab 1 a 1 b 135 15 7 15 17 p. 591 Quotient Property n 1a a n ,b 0 1b Ab n p. 591 2 12 A5 15 Simplified Form for Radicals A radical is in simplified form if the following three conditions are satisfied. 1. The radicand has no factor raised to a power greater than or equal to the index. 218x3 29x2 2x 29x2 12x 2. No fraction appears in the radical. © 2001 McGraw-Hill Companies 3. No radical appears in a denominator. Note: Satisfying the third condition may require rationalizing the denominator. 3x12x 5 15 15 A9 19 3 3 13 13 17x A 7x 17x 17x 17x 121x 249x 2 121x 7x p. 592 Continued 657 658 CHAPTER 8 RADICAL EXPRESSIONS DEFINITION /PROCEDURE EXAMPLE REFERENCE Operations on Radical Expressions Radical expressions may be combined only if they are similar, that is, if they have the same radicand with the same index. Similar radicals are combined by application of the distributive property. Multiplication To multiply two radical expressions, we use n n n 1 a 1 b 1 ab and simplify the product. If binomial expressions are involved, we use the distributive property or the FOIL method. Division To divide two radical expressions, rationalize the denominator by multiplying the numerator and denominator by the appropriate radical. If the divisor (the denominator) is a binomial, multiply the numerator and denominator by the conjugate of the denominator. Section 8.3 815 315 (8 3)15 1115 2118 412 219 2 4 12 219 12 4 12 2 3 12 4 12 612 412 (6 4)12 212 p. 601 13x 26x 218x 29x2 2x 29x2 12x 2 3 3x12x 12(5 18) 12 5 12 18 5 12 4 (3 12)(5 12) 15 312 512 2 13 212 5 12 512 5 18 18 12 116 512 4 p. 604 Note: 3 15 is the conjugate of 3 15. 2 2(3 15) 3 15 (3 15)(3 15) 2(3 15) 4 3 15 2 Solving Radical Equations Power Property of Equality If a b then an bn p. 607 Section 8.4 If 1x 1 5 then (1x 1)2 52 x 1 25 If an equation involves two radicals, rewrite the equation so that there is one radical on each side and then use the power property to solve it. p. 615 Given 1x 1x 7 7 1x 7 1x 7 x 49 141x 7 (x 7) x 56 x 141x 7 56 14 1x 7 4 1x 7 16 x 7 x9 p. 618 © 2001 McGraw-Hill Companies x 24 SUMMARY DEFINITION /PROCEDURE EXAMPLE REFERENCE Geometric and Other Applications of Radical Expressions Pythagorean Theorem Section 8.5 Given triangle c a 659 x 3 7 b x2 72 32 a2 b2 c2 x2 49 9 In a right triangle, the sum of the squares of the two sides is equal to the square of the hypotenuse. x 158 7.6 p. 627 Rational Exponents Section 8.6 Rational exponents are an alternate way of indicating roots. We use the following definition. If a is any real number and n is a positive integer (n 1), n a1n 1 a We restrict a so that a is nonnegative when n is even. We also define the following. For any real number a and positive integers m and n, with n 1, then n n amn (1 a)m 2 am 36 12 136 6 3 2713 127 3 5 24315 1243 3 1 1 125 5 2512 3 2723 (127)2 32 9 4 (a4b8)34 2(a4b8)3 4 2a12b24 a3b6 p. 635 Properties of Exponents The following five properties for exponents continue to hold for rational exponents. Product Rule am an amn x12 x13 x1213 x56 p. 638 am am n an x32 x32 12 x22 x x12 p. 638 (am)n am n (x13)5 x13 5 x53 p. 638 (ab)m ambm (2xy)12 212x12y12 p. 638 Quotient Rule Power Rule Product-Power Rule © 2001 McGraw-Hill Companies Quotient-Power Rule b a m am bm x13 3 2 (x13)2 32 x23 9 Complex Numbers p. 638 Section 8.7 The number i is defined as i 11 116 4i 18 2i12 Continued 660 CHAPTER 8 RADICAL EXPRESSIONS DEFINITION /PROCEDURE EXAMPLE REFERENCE Complex Numbers Section 8.7 Note that this means that i2 1 A complex number is any number that can be written in the form a bi in which a and b are real numbers and i 11 Addition and Subtraction For the complex numbers a bi and c di, (a bi) (c di) (a c) (b d )i p. 647 (2 3i) (3 5i) (2 3) (3 5)i 1 2i and (a bi) (c di) (a c) (b d )i (5 2i) (3 4i) (5 3) (2 (4))i 2 2i (a bi)(c di) (ac bd) (ad bc)i Note: It is generally easier to use the FOIL multiplication pattern and the definition of i, rather than to apply the above formula. Division To divide two complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator and write the result in standard form. (2 5i)(3 4i) 6 8i 15i 20i2 6 7i 20(1) 26 7i p. 649 3 2i (3 2i)(3 2i) 3 2i (3 2i)(3 2i) 9 6i 6i 4i2 9 4i2 9 12i 4(1) 9 4(1) 5 12 5 12i i 13 13 13 p. 651 © 2001 McGraw-Hill Companies Multiplication For the complex numbers a bi and c di, p. 648 Summary Exercises This summary exercise set is provided to give you practice with each of the objectives of the chapter. Each exercise is keyed to the appropriate chapter section. The answers are provided in the instructor’s manual that accompanies this text. Your instructor will provide guidelines on how to best use these exercises in your instructional program. [8.1] Evaluate each of the following roots over the set of real numbers. 1. 1121 2. 164 3 5. 164 6. 181 3 8 A 27 2 9. 28 4. 164 7. 9 A 16 3 8. 3. 181 4 Simplify each of the following expressions. Assume that all variables represent positive real numbers for all subsequent exercises in this exercise set. 10. 24x2 11. 2a4 13. 249w4z6 14. 2x9 3 16. 28r3s9 [8.2] 12. 236y2 3 15. 227b6 4 18. 232p5q15 17. 216x4y8 3 5 Use the product property to write each of the following expressions in simplified form. 19. 145 20. 175 22. 2108a3 23. 132 3 21. 260x2 3 24. 280w4z3 © 2001 McGraw-Hill Companies Use the quotient property to write each of the following expressions in simplified form. 25. 9 A 16 26. 7 A 36 27. y4 B 49 28. 2x A9 29. 5 A 16x2 30. 5a2 B 27 [8.3] 3 Simplify each of the following expressions if necessary. Then add or subtract as indicated. 3 3 31. 7110 4110 32. 513x 213x 33. 712x 3 12x 34. 8110 3110 2110 35. 172 150 36. 154 124 37. 917 2163 38. 120 145 21125 39. 2116 3154 3 3 661 662 CHAPTER 8 RADICAL EXPRESSIONS 3 40. 227w w112w 43. 172x [8.3] x A2 3 3 5 2 41. 2128a 6a22a 3 42. 120 3 a A9 4 44. 281a a 45. 3 15 115 1 3 115 Multiply and simplify each of the following expressions. 3 3 46. 13x 17y 2 47. 26x 118 2 2 48. 24a b 2ab 49. 15(13 2) 50. 16(18 12) 51. 1a(15a 1125a) 52. (13 5)(13 7) 53. (17 12)(17 13) 54. (15 2)(15 2) 55. (17 13)(17 13) 56. (2 13)2 57. (15 12)2 Rationalize the denominator, and write each of the following expressions in simplified form. 58. 3 A7 59. 3 61. A a2 112 1x 60. 3 2 62. 3 13x 3 110a 15b 63. 2x2 3 2y5 Divide and simplify each of the following expressions. 64. 1 3 12 65. 11 5 13 66. 15 2 15 2 67. 1x 3 1x 3 Solve each of the following equations. Be sure to check your solutions. 68. 1x 5 4 69. 13x 2 2 5 70. 1y 7 y 5 71. 12x 1 x 8 3 3 72. 15x 2 3 2 73. 2x 2 3 0 74. 1z 7 1 1z 75. 14x 5 1x 1 3 Solve each of the following equations for the indicated variable. 2 2 76. r 2x y for x 77. t 2p l A 10 for l © 2001 McGraw-Hill Companies [8.4] SUMMARY EXERCISES [8.5] 663 Use the Pythagorean theorem to find the length x in each triangle. Write answers to the nearest tenth. 78. 79. x 14 in. 10 cm x 8 in. 9 cm 80. Find the width of a rectangle whose diagonal is 12 ft and whose length is 6 ft. 81. If a window is 3 ft on its shorter side, how high should it be, to the nearest tenth of a foot, to be a golden rectangle? 82. The base of a ladder that is 16 ft long is placed 4 ft from a wall of a building. How high up the wall is the top of the ladder? 83. How long must a ladder be to reach 8 m up the side of a building when the base of the ladder is 2 m from the building? [8.6] Evaluate each of the following expressions. 84. 4912 85. 10012 86. (27)13 87. 1614 88. 6423 89. 2532 91. 4912 92. 8134 90. 9 4 32 Use the properties of exponents to simplify each of the following expressions. 93. x32 x52 © 2001 McGraw-Hill Companies 95. 94. b23 b32 r85 r35 96. 98. (y43)6 97. (x35)23 100. (16x13 y23)34 99. (x45y32)10 101. x2y16 x4y a54 a12 3 102. 27y3z6 x3 13 664 CHAPTER 8 RADICAL EXPRESSIONS Write each of the following expressions in radical form. 103. x34 104. (w2z)25 105. 3a23 106. (3a)23 Write each of the following expressions, using rational exponents, and simplify when necessary. 5 4 108. 216w 3 110. 216a8b16 107. 17x 109. 227p3q9 [8.7] 4 Write each of the following roots as a multiple of i. Simplify your result. 111. 149 112. 113 113. 160 Perform the indicated operations. 114. (2 3i) (3 5i) 115. (7 3i) (3 2i) 116. (5 3i) (2 5i) 117. (4 2i) (1 3i) Find each of the following products. 118. 4i(7 2i) 119. (5 2i)(3 4i) 120. (3 4i)2 121. (2 3i)(2 3i) 122. 5 15i 5i 123. 10 3 4i 124. 3 2i 3 2i 125. 5 10i 2i © 2001 McGraw-Hill Companies Find each of the following quotients, and write your answer in standard form.