Multiplying and Dividing Radicals

9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated the first property for radicals: 1ab 1a 1b when a and b are any positive real numbers That property has been used to simplify radical expressions up to this point. Suppose now that we want to find a product, such as 1 15. We can use our first radical rule in the opposite manner. NOTE The product of square roots is equal to the square root of the product of the radicands. 1a 1b 1ab so 1 15 1 5 115 We may have to simplify after multiplying, as Example 1 illustrates. Example 1 Simplifying Radical Expressions Multiply then simplify each expression. (a) 15 110 15 10 150 125 2 512 (b) 112 16 112 6 172 16 2 16 12 612 An alternative approach would be to simplify 112 first. 112 16 21 16 2118 219 2 219 12 2 12 612 (c) 110x 12x 220x 2 24x 2 5 24x 2 15 2x15 72

724 CHAPTER 9 EXPONENTS AND RADICALS CHECK YOURSELF 1 Simplify. (a) 1 16 (b) 1 118 (c) 18a 1a If coefficients are involved in a product, we can use the commutative and associative properties to change the order and grouping of the factors. This is illustrated in Example 2. Example 2 Multiplying Radical Expressions NOTE In practice, it is not necessary to show the intermediate steps. (215)(16) (2 )(15 16) 615 6 610 CHECK YOURSELF 2 Multiply ( 17)( 51). The distributive property can also be applied in multiplying radical expressions. Consider the following. Example Multiplying Radical Expressions (a) 1(12 1) 1 12 1 1 16 The distributive property Multiply the radicals. (b) 15(216 1) 15 216 15 1 2 15 16 15 1 210 115 The distributive property The commutative property CHECK YOURSELF (a) 15(16 15) (b) 1(215 12) The FOIL pattern we used for multiplying binomials in Section.4 can also be applied in multiplying radical expressions. This is shown in Example 4.

MULTIPLYING AND DIVIDING RADICALS SECTION 9.4 725 Example 4 Multiplying Radical Expressions (a) (1 2)(1 5) 1 1 51 21 2 5 51 21 10 Combine like terms. 1 71 CAUTION NOTE You can use the pattern (a b)(a b) a 2 b 2, where a 17 and b 2, for the same result. 17 2 and 17 2 are called conjugates of each other. Note that their product is the rational number. The product of conjugates will always be rational. Be Careful! This result cannot be further simplified: 1 and 71 are not like terms. (b) (c) (17 2)(17 2) 17 17 217 217 4 7 4 (1 5) 2 (1 5)(1 5) 1 1 51 51 5 5 51 51 25 28 101 CHECK YOURSELF 4 (a) (15 )(15 2) (b) (1 4)(1 4) (c) (12 ) 2 We can also use our second property for radicals in the opposite manner. NOTE The quotient of square roots is equal to the square root of the quotient of the radicands. 1a 1b a A b One use of this property to divide radical expressions is illustrated in Example 5. Example 5 Simplifying Radical Expressions Simplify. NOTE The clue to recognizing when to use this approach is in noting that 48 is divisible by. (a) (b) 148 1 1200 12 A 48 116 4 A 200 2 1100 10 2125x 2 125x 2 (c) 225x 2 5x 15 A 5 There is one final quotient form that you may encounter in simplifying expressions, and it will be extremely important in our work with quadratic equations in the next chapter. This form is shown in Example 6.

726 CHAPTER 9 EXPONENTS AND RADICALS CHECK YOURSELF 5 Simplify. (a) 175 1 (b) 281s2 19 Example 6 Simplifying Radical Expressions Simplify the expression 172 CAUTION First, we must simplify the radical in the numerator. Be Careful! Students are sometimes tempted to write 612 1 612 This is not correct. We must divide both terms of the numerator by the common factor. 172 16 2 16 12 (1 212) 612 1 212 Use Property 1 to simplify 172. Factor the numerator then divide by the common factor. CHECK YOURSELF 6 Simplify 15 175. 5 CHECK YOURSELF ANSWERS 1. (a) 12; (b) 16; (c) 2a16 2. 15121. (a) 10 5; (b) 2115 16 4. (a) 1 15; (b) 1; (c) 11 612 5. (a) 5; (b) s 6. 1

Name 9.4 Exercises Section Date Perform the indicated multiplication. Then simplify each radical expression. 1. 17 15 2. 1 17 ANSWERS 1.. 15 111 4. 5. 1 110m 6. 7. 12x 115 8. 9. 1 1 7 12 10. 11. 1 112 12. 11 15 17a 11 117 12b 15 17 1 17 17 2.. 4. 5. 6. 7. 8. 9. 10. 11. 12. 1. 1. 110 110 14. 15. 118 16 16. 17. 12x 16x 18. 15 115 18 110 1a 115a 14. 15. 16. 17. 18. 19. 20. 19. 21 17 20. 12 15 21. (1)(517) 22. (2 15)( 111) 21. 22. 727

ANSWERS 2. 24. 25. 26. 2. (15)(2 110) 24. 25. 15(12 15) 26. (41)(16) 1(15 1) 27. 28. 29. 27. 1(215 1) 28. 17(21 17) 0. 1. 29. (1 5)(1 ) 0. (15 2)(15 1) 2.. 4. 5. 6. 7. 1. (15 1)(15 ) 2.. (15 2)(15 2) 4. 5. (110 5)(110 5) 6. (12 )(12 7) (17 5)(17 5) (111 )(111 ) 8. 9. 40. 7. (1x )(1x ) 8. (1a 4)(1a 4) 41. 42. 9. (1 2) 2 40. (15 ) 2 4. 44. 41. (1y 5) 2 42. (1x 4) 2 45. 46. Perform the indicated division. Rationalize the denominator if necessary. Then simplify each radical expression. 198 4. 44. 12 272a 2 45. 46. 12 1108 1 248m 2 1 728

ANSWERS 4 148 47. 48. 4 12 1108 6 47. 48. 5 1175 49. 5 50. 8 1512 51. 4 52. 6 118 5. 5. 15 175 55. 5 56. 18 1567 9 9 1108 6 120 2 8 148 4 49. 50. 51. 52. 5. 54. 55. 57. Area of a rectangle. Find the area of the rectangle shown in the figure. 11 56. 57. 58. 59. 58. Area of a rectangle. Find the area of the rectangle shown in the figure. 5 60. 5 61. 59. Complete this statement: 12 15 110 because.... 60. Explain why 21 51 71 but 71 15 1018. 61. When you look out over an unobstructed landscape or seascape, the distance to the visible horizon depends on your height above the ground. The equation d A 2 h is a good estimate of this, in which d distance to horizon in miles and h height of viewer above the ground. Work with a partner to make a chart of distances to the horizon given different elevations. Use the actual heights of tall buildings or 729

ANSWERS a. b. c. prominent landmarks in your area. The local library should have a list of these. Be sure to consider the view to the horizon you get when flying in a plane. What would your elevation have to be to see from one side of your city or town to the other? From one side of your state or county to the other? d. e. f. Getting Ready for Section 9.5 [Section 9.1] Evaluate the following. Round your answer to the nearest thousandth. (a) (c) (e) 116 1121 127 (b) (d) (f) 149 112 198 Answers 1. 15. 155 5. 10m 7. 10x 9. 142 11. 6 1. 10 15. 61 17. 2x1 19. 2121 21. 15121 2. 012 25. 110 5 27. 2115 9 29. 18 81 1. 2 215. 1 5. 15 7. x 9 9. 7 41 41. y 101y 25 4. 7 45. 6a 47. 1 1 49. 1 17 51. 2 412 5. 2 12 55. 1 57. 1 59. 61. a. 4 b. 7 c. 11 d..464 e. 5.196 f. 9.899 70