Section 9.2 Multiplication Properties of Radicals In Exercises 27-46, place each of the 27.

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1 Section 9.2 Multiplication Properties of Radicals Exercises 1. Use a calculator to first approximate 5 2. On the same screen, approximate 10. Report the results on your homework paper. 2. Use a calculator to first approximate On the same screen, approximate 70. Report the results on your homework paper Use a calculator to first approximate On the same screen, approximate 33. Report the results on your homework paper. 4. Use a calculator to first approximate On the same screen, approximate 65. Report the results on your homework paper. In Exercises 5-20, place each of the radical expressions in simple radical form. As in Example 3 in the narrative, check your result with your calculator In Exercises 21-26, use prime factorization (as in Examples 10 and 11 in the narrative) to assist you in placing the given radical expression in simple radical form. Check your result with your calculator In Exercises 27-46, place each of the given radical expressions in simple radical form. Make no assumptions about the sign of the variables. Variables can either represent positive or negative numbers (6x 11) 4 1 Copyrighted material. See:

2 902 Chapter 9 Radical Functions h 8 25f 2 25j 8 16m 2 25a 2 (7x + 5) 12 9w 10 25x 2 50x x 2 42x x x f 14 (3x + 6) 12 (9x 8) 12 for x = Given that x < 0, place the radical expression 27x 12 in simple radical form. Check your solution on your calculator for x = Given that x < 0, place the radical expression 44x 10 in simple radical form. Check your solution on your calculator for x = 2. In Exercises 51-54, follow the lead of Example 17 in the narrative to simplify the given radical expression and check your result with your graphing calculator. 51. Given that x < 4, place the radical expression x 2 8x + 16 in simple radical form. Use a graphing calculator to show that the graphs of the original expression and your simple radical form agree for all values of x such that x < x x + 9 4e 2 4p 10 25x 12 25q 6 16h Given that x < 0, place the radical expression 32x 6 in simple radical form. Check your solution on your calculator for x = Given that x < 0, place the radical expression 54x 8 in simple radical form. Check your solution on your calculator 52. Given that x 2, place the radical expression x 2 + 4x + 4 in simple radical form. Use a graphing calculator to show that the graphs of the original expression and your simple radical form agree for all values of x such that x Given that x 5, place the radical expression x 2 10x + 25 in simple radical form. Use a graphing calculator to show that the graphs of the original expression and your simple radical form agree for all values of x such that x Given that x < 1, place the radical expression x 2 + 2x + 1 in simple radical form. Use a graphing calculator to show that the graphs of the original expression and your simple radical form agree for all values of x such that x < 1.

3 Section 9.2 Multiplication Properties of Radicals 903 In Exercises 55-72, place each radical expression in simple radical form. Assume that all variables represent positive numbers. In Exercises 73-80, place each given radical expression in simple radical form. Assume that all variables represent positive numbers d f 5 8f k s 3 243s x x k 7 32k x 2 30x n 9 8n j e 9 8e j n 9 125n m z 5 27z e t 7 27t c z h b s e p d q w 7

4 Chapter 9 Radical Functions 9.2 Solutions 1. Note that 5 2 = Note that 3 11 = = 32 2 = = = 42 7 = = = 62 3 = = = 52 2 = = = 72 5 = = = 72 2 = = 7 2

5 Section 9.2 Multiplication Properties of Radicals = 32 5 = = = 22 6 = = Note that = 9, which is divisible by 9. Thus, 2016 is divisible by 9. Indeed, 2016 = The last two digits of 224 are 24, which is divisible by 4. Thus, 224 is divisible by 4. Indeed, 224 = Continue to primes = = (3 3) (4 56) = = Factor out a perfect square (exponents must be divisible by 2) = = = = Checking, 23. Money! Anything that ends in 00, 25, 50, or 75 is divisible by 25. Indeed, = Further, = 18, so 567 is divisible by 9; i.e., 567 = Continuing to primes, = = = Factor our a perfect square (exponents divisible by 2) = = = = 45 7

6 Chapter 9 Radical Functions Checking, 25. Money! Anything that ends in 00, 25, 50, or 75 is divisible by 25. Indeed, = Continuing to primes, = = Factor out a perfect square = = = = Checking, 27. (6x 11) 4 = ((6x 11) 2 ) 2 = (6x 11) 2 However, (6x 11) 2 is already nonnegative, so the absolute value bars are unnecessary. Hence, (6x 11) 4 = (6x 11) f 2 = 25 f 2 = 5 f Because f can be any real number, we cannot remove the absolute value bars without more information m 2 = 4 2 m 2 = 4 2 m 2 = 4 m Since the index on the radical is even and, after simplification, the variable is raised to an odd power, absolute value signs around the simplified variable are necessary.

7 Section 9.2 Multiplication Properties of Radicals 33. (7x + 5) 12 = ((7x + 5) 6 ) 2 = (7x + 5) 6 However, (7x + 5) 6 is already nonnegative, so absolute value signs are unnecessary. (7x + 5) 12 = (7x + 5) x 2 50x + 25 = (5x 5) 2 = 5x 5 Because x can be any real number, the absolute value signs around the simplified binomial are necessary x x + 81 = (5x + 9) 2 = 5x + 9 Because x can be any real number, the absolute value signs around the simplified binomial are necessary. 39. (3x + 6) 12 = ((3x + 6) 6 ) 2 = (3x + 6) 6 However, the expression (3x + 6) 6 is already nonnegative, so the absolute value bars are unnecessary. (3x + 6) 12 = (3x + 6) x x + 9 = (6x + 3) 2 = 6x + 3 Because x can be any real number, the absolute value signs around the simplified binomial are necessary p 10 = 4 (p 5 ) 2 = 2 p 5 Now, we can use the multiplicative property of absolute values and write 2 p 5 = 2 p 4 p = 2p 4 p. Since p can be any real number, absolute value signs around the simplified variable are necessary.

8 Chapter 9 Radical Functions q 6 = 25 (q 3 ) 2 = 5 q 3 Now, we can use the multiplicative property of absolute values and write 5 q 3 = 5 q 2 q = 5q 2 q. Because q can be any real number, absolute value signs around the simplified variable are necessary. 47. Factor out a perfect square. 32x 6 = 16x 6 2 = 16 x 6 2 = 4 x 3 2 However, x 3 = x 2 x = x 2 x, since x 2 0. Thus, 32x 6 = 4x 2 x 2. If x < 0, then x = x and 32x 6 = 4x 2 ( x) 2 = 4x 3 2. Checking with x = Factor out a perfect square. 27x 12 = 9x 12 3 = 9 x 12 3 = 3 x 6 3. However, x 6 = x 6 since x 6 0. Thus, 27x 12 = 3x 6 3. Checking with x = 2.

9 Section 9.2 Multiplication Properties of Radicals 51. Factor the perfect square trinomial. x 2 8x + 16 = (x 4) 2 = x 4 If x < 4, or equivalently, if x 4 < 0, then x 4 = (x 4). Thus, x 2 8x + 16 = x + 4. In (b), we ve drawn the graph of y = x 2 8x In (d), we ve drawn the graph of y = x + 4. Note that the graphs in (b) and (d) agree when x < 4, lending credence to the fact that x 2 8x + 16 = x + 4 when x < 4. (a) (b) (c) (d) 53. Factor the perfect square trinomial. x 2 10x + 25 = (x 5) 2 = x 5 If x 5, or equivalently, x 5 0, then x 5 = x 5. Hence, x 2 10x + 25 = x 5. (1) In (b), we ve drawn the graph of y = x 2 10x In (d), we ve drawn the graph of y = x 5. Note that the graphs in (b) and (d) agree when x 5, lending credence to the fact that x 2 10x + 25 = x 5 when x 5. (a) (b) (c) (d) 55. 9d 13 = 9 d 12 d = 3d 6 d x x + 16 = (5x + 4) 2 = 5x + 4

10 Chapter 9 Radical Functions 59. 4j 11 = 4 j 10 j = 2j 5 j m 2 = 25 m 2 = 5m 63. 4c 5 = 4c 4 c = 2c 2 c h 10 = 25 h 10 = 5h s 7 = 9 s 6 s = 3s 3 s 69. 4p 8 = 4 p 8 = 2p q 10 = 9 q 10 = 3q f 5 8f 3 = 2 8 f 5 f 3 = 16f 8 = 16 (f 4 ) 2 = 4f k 7 32k 3 = 2 32 k 7 k 3 = 64k 10 = 64 (k 5 ) 2 = 8k e 9 8e 3 = 2 8 e 9 e 3 = 16e 12 = 16 (e 6 ) 2 = 4e z 5 27z 3 = 3 27 z 5 z 3 = 81z 8 = 81 (z 4 ) 2 = 9z 4