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Chapter 7 488 Radicals and Complex Numbers section 7.3 Objectives 1. Multiplication and Division Properties of Radicals 2. Simplifying Radicals by Using the Multiplication Property of Radicals 3. Simplifying Radicals by Using the Division Property of Radicals Simplifying Radical Expressions 1. Multiplication and Division Properties of Radicals You may have already noticed certain properties of radicals involving a product or quotient. Multiplication and Division Properties of Radicals n n Let a and b represent real numbers such that 1 a and 1 b are both real. Then n n n 1. 1 ab 1 a 1 b Multiplication property of radicals n 2. Concept Connections Multiply or divide the radicals. 1. 15 16 2. a 1a n n Ab 1b b0 Division property of radicals Properties 1 and 2 follow from the properties of rational exponents. 1ab 1ab2 1n a1nb1n 1a 1b n n n a a 1n a1n 1a a b 1n n Ab b b 1b n n 110 15 The multiplication and division properties of radicals indicate that a product or quotient within a radicand can be written as a product or quotient of radicals, provided the roots are real numbers. For example: 1144 116 19 25 125 A 36 136 The reverse process is also true. A product or quotient of radicals can be written as a single radical provided the roots are real numbers and they have the same indices. 13 112 136 3 18 3 2125 8 3 A 125 In algebra it is customary to simplify radical expressions as much as possible. Simplified Form of a Radical Consider any radical expression where the radicand is written as a product of prime factors. The expression is in simplified form if all the following conditions are met: 1. The radicand has no factor raised to a power greater than or equal to the index. 2. The radicand does not contain a fraction. 3. There are no radicals in the denominator of a fraction. Answers 1. 130 2. 12 Section 7.3 Simplifying Radical Expressions 489 2. Simplifying Radicals by Using the Multiplication Property of Radicals The expression 2x2 is not simplified because it fails condition 1. Because x2 is a perfect square, 2x2 is easily simplified: 2x2 x for x 0 However, how is an expression such as 2x9 simplified? This and many other radical expressions are simplified by using the multiplication property of radicals. The following examples illustrate how nth powers can be removed from the radicands of nth roots. example 1 Using the Multiplication Property to Simplify a Radical Expression Skill Practice Use the multiplication property of radicals to simplify the expression 2x . Assume x 0. 9 Simplify the expressions. Assume all variables represent positive real numbers. 3. 2a3 4. 2y 31 Solution: The expression 2x9 is equivalent to 2x8 x. Applying the multiplication property of radicals, we have 2x9 2x8 x 2x8 2x Apply the multiplication property of radicals. Note that x8 is a perfect square because x8 1x4 2 2. x4 1x Simplify. In Example 1, the expression x9 is not a perfect square. Therefore, to simplify 2x , it was necessary to write the expression as the product of the largest perfect square and a remaining or “left-over” factor: 2x9 2x8 x. This process also applies to simplifying nth roots, as shown in Example 2. 9 example 2 Using the Multiplication Property to Simplify a Radical Expression Use the multiplication property of radicals to simplify each expression. Assume all variables represent positive real numbers. 4 3 b. 2 w7z9 a. 2b7 Skill Practice Simplify the expressions. Assume all variables represent positive real numbers. 4 25 5. 2 v 3 8 12 6. 2 pq Solution: The goal is to rewrite each radicand as the product of the largest perfect square (perfect cube, perfect fourth power, and so on) and a left-over factor. 4 4 a. 2b7 2b4 b3 4 b4 is the largest perfect fourth power in the radicand. 4 2 b4 2 b3 Apply the multiplication property of radicals. Answers 4 3 b2b Simplify. 3. a1a 4 5. v 6 1v 4. y15 1y 3 6. p 2q 4 2p 2 490 Chapter 7 Radicals and Complex Numbers 3 3 b. 2w7z9 2 1w6z9 2 1w2 3 3 2 w6z9 2 w 2 3 3 w z 1w w6z9 is the largest perfect cube in the radicand. Apply the multiplication property of radicals. Simplify. Each expression in Example 2 involves a radicand that is a product of variable factors. If a numerical factor is present, sometimes it is necessary to factor the coefficient before simplifying the radical. Skill Practice Simplify the radicals. Assume all variables represent positive real numbers. 7. 224 8. 5218 4 9. 232a10b19 example 3 Using the Multiplication Property to Simplify Radicals Use the multiplication property of radicals to simplify the expressions. Assume all variables represent positive real numbers. a. 156 b. 6250 Solution: a. 256 223 7 Factor the radicand. 2122 2 12 72 Avoiding Mistakes: The multiplication property of radicals allows us to simplify a product of factors within a radical. For example: 2x 2y 2 2x 2 2y 2 xy However, this rule does not apply to terms that are added or subtracted within the radical. For example: 2x 2 y 2 and 2x 2 y 2 cannot be simplified. 3 c. 240x3y5z7 22 is the largest perfect square in the radicand. 222 22 7 Apply the multiplication property of radicals. 2114 Simplify. Calculator Connections A calculator can be used to support the solution to Example 3(a). The decimal approximation for 156 and 2114 agree for the first 10 digits. This in itself does not make 156 2114. It is the multiplication of property of radicals that guarantees that the expressions are equal. b. 6250 6 22 52 Factor the radicand. 6 25 22 Apply the multiplication property of radicals. 6 5 22 Simplify. 3022 Simplify. 2 3 c. 240x3y5z7 3 22 5x y z 3 3 5 7 2 123x3y3z6 2 15y2z2 3 3 3 3 3 6 3 2 2xyz 2 5y2z Answers 7. 2 16 8. 15 12 4 9. 2a2b4 22a2b3 2 56 2 28 2 14 7 3 2xyz2 2 5y2z Factor the radicand. 23x3y3z6 is the largest perfect cube. Apply the multiplication property of radicals. Simplify. 2 50 5 25 5 2 40 2 20 2 10 5 Section 7.3 Simplifying Radical Expressions 3. Simplifying Radicals by Using the Division Property of Radicals The division property of radicals indicates that a radical of a quotient can be written as the quotient of the radicals and vice versa, provided all roots are real numbers. example 4 Using the Division Property to Simplify Radicals Use the division property of radicals to simplify the expressions. Assume all variables represent positive real numbers. a. a7 B a3 b. 3 1 3 3 181 c. 7150 15 d. 2c5 B 32cd8 4 Solution: a. a7 B a3 The radicand contains a fraction. However, the fraction can be reduced to lowest terms. Skill Practice Use the division property of radicals to simplify the expressions. Assume all variables represent positive real numbers. 10. v 21 B v5 11. 12. 22300 30 13. 2a4 a2 b. c. Simplify the radical. 3 1 3 The expression has a radical in the denominator. 3 1 81 3 A 81 1 B27 Reduce to lowest terms. 1 3 Simplify. 3 3 7150 15 Because the radicands have a common factor, write the expression as a single radical (division property of radicals). Simplify 150. 7252 2 15 52 is the largest perfect square in the radicand. 7252 22 15 Multiplication property of radicals 7 512 15 Simplify the radicals. 1 7 512 15 Reduce to lowest terms. 3 712 3 Answers 10. v 8 2 13 12. 3 11. 2 3x 5 13. 6 y 5 264 5 2 2 3 54x17y B 2x 2y19 491 492 Chapter 7 Radicals and Complex Numbers d. 2c5 B 32cd8 4 section 7.3 The radicand contains a fraction. c4 B 16d8 4 Simplify the factors in the radicand. 4 4 2 c Apply the division property of radicals. 4 2 16d8 c 2d2 Simplify. Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos For the exercises in this set, assume that all variables represent positive real numbers unless otherwise stated. Study Skills Exercise 1. When writing a radical expression, be sure that you understand the difference between an exponent on a coefficient and an index to a radical. Write an algebraic expression for each of the following. a. x cubed times the square root of y b. x times the cube root of y Review Exercises For Exercises 2–4, simplify the expression. Write the answer with positive exponents only. 2. 1a2b4 2 1 2 a a b b3 3. a p4 q 6 b 12 1p3q2 2 4. 1x13y56 2 6 5. Write x4 7 in radical notation. 6. Write y25 in radical notation. 7. Write 2y9 by using rational exponents. 3 8. Write 2x2 by using rational exponents. Objective 2: Simplifying Radicals by Using the Multiplication Property of Radicals For Exercises 9–30, simplify the radicals. (See Examples 1–3.) 9. 2x11 10. 2p15 3 11. 2q7 3 12. 2r17 13. 2a5b4 14. 2c9d6 4 15. 2x8y13 4 16. 2p16q17 Section 7.3 Simplifying Radical Expressions 17. 128 18. 163 19. 180 20. 1108 3 21. 254 3 22. 2250 23. 225ab3 24. 264m5n20 25. 218a6b3 26. 272m5n2 3 27. 216x6yz3 3 28. 2192a6bc2 4 29. 280w4z7 4 30. 232p8qr5 493 Objective 3: Simplifying Radicals by Using the Division Property of Radicals For Exercises 31–42, simplify the radicals. (See Example 4.) 31. x3 Bx 32. y5 By 33. 35. 50 B2 36. 98 B2 37. 39. 3 51 16 6 40. 7118 9 41. 2p7 2p3 3 1 3 3 1 24 3 51 72 12 34. 38. 42. 2q11 2q5 3 1 3 3 1 81 3 31 250 10 Mixed Exercises For Exercises 43–58, simplify the radicals. 43. 5118 44. 2124 45. 6175 46. 818 47. 225x4y3 48. 2125p3q2 3 49. 227x2y3z4 3 50. 2108a3bc2 51. 55. 16a2b B 2a2b4 3 250x3y 29y 4 52. 3s2t4 B 10,000 4 53. 32x B y10 5 54. 3 16j 3 B k3 3 56. 227a4 3 28a 57. 223a14b8c31d22 58. 275u12v20w65x80 For Exercises 59–62, write a mathematical expression for the English phrase and simplify. 59. The quotient of 1 and the cube root of w6 60. The principal square root of the quotient of h and 49 61. The principal square root of the quantity k raised to the third power 62. The cube root of 2x4 Chapter 7 494 Radicals and Complex Numbers For Exercises 63–66, find the third side of the right triangle. Write your answer as a radical and simplify. 63. 64. 10 ft ? 6 in. ? 12 in. 8 ft 65. ? 66. 18 m 3 cm 12 m 7 cm ? 67. On a baseball diamond, the bases are 90 ft apart. Find the exact distance from home plate to second base. Then round to the nearest tenth of a foot. 68. Linda is at the beach flying a kite. The kite is directly over a sand castle 60 ft away from Linda. If 100 ft of kite string is out (ignoring any sag in the string), how high is the kite? (Assume that Linda is 5 ft tall.) See figure. 2nd base 100 ft 90 ft 60 ft 5 ft 90 ft Home plate Expanding Your Skills 69. Tom has to travel from town A to town C across a small mountain range. He can travel one of two routes. He can travel on a four-lane highway from A to B and then from B to C at an average speed of 55 mph. Or he can travel on a two-lane road directly from town A to town C, but his average speed will be only 35 mph. If Tom is in a hurry, which route will take him to town C faster? B C 40 mi 70. One side of a rectangular pasture is 80 ft in length. The diagonal distance is 110 yd. If fencing costs $3.29 per foot, how much will it cost to fence the pasture? 110 yd 80 ft 50 mi A Section 7.4 chapter 7 3 1. 164 1 A 16 4. 5. 254x w 7. For Exercises 12–19, simplify the exponential expressions. Leave no negative exponents in your final answer. 12. 823 13. 1634 2. 181 4 3 495 midchapter review In the following exercises, assume all variables represent positive real numbers. For Exercises 1–10, simplify the expressions. 3. Addition and Subtraction of Radicals 2 27 A 125 3 6. 220t u v 4 2 3 7 50x3y5 B x 8. 16w5z10 B 2w2 3 14. a 1 12 b 25 15. a 4 1 2 b 25 16. a s2t4 1 2 b r6 17. a p2q3 1 2 b r5 18. 1u2v3 2 16 1u3v6 2 13 19. 1x1 2y1 4 2 1x12y1 3 2 20. Explain how to simplify 12523. 2 2 3 9. 2 14r 32 3 10. ab B 1a b2 2 5 51 11. Explain how to simplify 2 x . 21. Rewrite the expressions, using rational exponents. 3 a. 2 7x2 b. 221a 22. Rewrite the expressions in radical notation. a. x23 section 7.4 Addition and Subtraction of Radicals Definition of Like Radicals Two radical terms are said to be like radicals if they have the same index and the same radicand. The following are pairs of like radicals: same index and 3a15 same radicand Objectives 1. Definition of Like Radicals 2. Addition and Subtraction of Radicals 1. Definition of Like Radicals 7a15 b. 1c14 2 3 Indices and radicands are the same.