UNDERSTAND PRACTICE

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Practice Tutorial

PRACTICE & PROBLEM SOLVING

What is the reduced radical form of each expression? SEE EXAMPLE 1

19. ( 3x 1 __ 2 ) ( 4x

2 __ 3 ) 20. 2b

1 __ 2 ( 3b

1 __ 2 c

1 __ 3 )

2

21. ( x 1 __ 2 ∙ x

5 __ 12

)

4

÷ x 2 __ 3 22. ( 16c 14 ______

81d 18 )

1 __ 2

What is the reduced radical form of each expression? SEE EXAMPLE 2

23. 3 √ ________

250y 2 z 4 24. 4 √ __________

256v 7 w 12

25. √

______

48x 3 ______ 3xy 2

26. √

______

56x 5 y 5

______ 7xy

27. 3 √ ______

216m 28. 3 √

________

250f 7 g 3

_______ 2f 2 g

What is the reduced radical form of each expression? SEE EXAMPLE 3

29. √ _____

x 5 y 5 ∙ 3 √ ______

2 x 7 y 6 30. 3 √ _____

18n 2 ____ 24n

31. 3 √ ____

3x 2 ∙ 3 √ ___

x 2 ∙ 3 √ ____

9 x 3 32. √ _____

162a _____ 6a 3

33. 5 √ _____

2pq 6 ∙ 2 √ _____

2 p 3 q 34. 3 √

___

x 2 ___ 9y

35. 3 √ __

6 ∙ 3 √ ___

16 36. 4 √ ___

2 ___ 5x

What is the reduced radical form of each expression? SEE EXAMPLE 4

37. 4 3 √ ___

81 − 2 3 √ ___

72 − 3 √ ___

24 38. 6 √ _____

45y 2 − 4 √ _____

20y 2

39. 3 √ ___

12 − √ ___

54 + 7 √ ___

75 40. √ ____

32ℎ + 4 √ ____

98ℎ − 3 √ ____

50ℎ

Multiply. SEE EXAMPLE 5

41. (3 √ __

p − √ __

5 )( √ __

p + 5 √ __

5 ) 42. (4m − √ __

3 )(4m − √ __

3 )

43. (3 √ __

2 + 8)(3 √ __

2 − 8) 44. 3 √ __

3 (5 3 √ __

9 − 4)

What is the reduced radical form of each expression? SEE EXAMPLE 6

45. 4 ______ 1 − √

__ 3 46. 20 ______

3 + √ __

2

47. 3 + √ __

8 ______ 2 − 2 √

__ 8 48. −2x ______

3 + √ __

x

13. Model With Mathematics In the expression PV 4 __ 3 ,

P represents the pressure and V represents the volume of a sample of a gas. Evaluate the expression for P = 7 and V = 8 .

14. Reason Describe the possible values of k such that √

___ 32 + √

__ k can be rewritten as a single

term.

15. Error Analysis Explain why the following work is incorrect. Find the correct answer.

✗

12

125 4 – 5 = 5(4) – 5 5

= 15

12= 20 – 25

16. Communicate Precisely Discuss the advantages and disadvantages of first rewriting √

___ 27 + √

___ 48 + √

____ 147 in order to estimate its

decimal value.

17. Higher Order Thinking Write √ __

4 __ 5 in two different ways, one where the numerator is simplified and another where the denominator is rationalized.

18. Construct Arguments Justify each step used in simplifying the expression below.

=

=14a

= √a

=

153

4a2–

34

15

a2

a155

4a

4

LESSON 5-2 Properties of Exponents and Radicals 253

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PRACTICE & PROBLEM SOLVING

APPLY ASSESSMENT PRACTICE

Mixed Review Available Online

Practice Tutorial

49. Model With Mathematics A triangular swimming area is marked off by a rope.

a. If a woman swims around the perimeter of the swimming area, how far will she swim?

b. What is the area of the roped off section?

200 6 m √ 400 m

200 m 200 3 m √

200 3 m

√

50. Use Structure The interest rate r required to increase your investment p to the amount a

in m months is found by r = ( a __ p ) 1 __ m

− 1. What interest rate would be required to increase

your investment of $3,600 to $6,400 over 7 months? Round your answer to the nearest tenth of a percent.

51. Use Structure The length of a rectangle is (2 + √

__ 5 )y . The width is (4 + 3 √

__ 5 )z . What is the

area of the rectangle?

(2+√5 )y(4+3√5 )z

52. Model With Mathematics A rectangular boardroom table is √

____ 440 ft by √

___ 20 ft. Find its

area.

53. Aaron is rewriting 1 + √ __

3 ______ 5 − √

__ 3 into reduced radical

form. Determine if Aaron would have written the steps below to show his work. Select Yes or No.

Yes No

❑

❑

❑

❑❑

❑

❑

❑

❑ ❑

6 + 4√3 – 325 + 9

5 + 6√3 + 325 – 3

5 + √3 + 5√3 +√925 + 5√3 – 5√3 –√94 + 3√3

11

8 + 6√328

54. SAT/ACT Which expression cannot be rewritten as −10?

Ⓐ √ ___

25 ∙ 3 √ ___

−8 Ⓑ 3 √ _____

−125 ∙ 4 √ ___

16

Ⓒ − 3 √ _____

1,000 Ⓓ − √ ___

25 ∙ 5 √ ____

−32

Ⓔ √ __

4 ∙ − 3 √ ____

125

55. Performance Task The volume of a sphere of radius r is V = 4 __

3 πr 3 .

Part A Use the formula to find r in terms of V. Rationalize the denominator.

Part B A snowman is made using three spherical snowballs. The top snowball for the head has a volume of 500 in. 3 . What is the diameter of the top snowball?

V = 1,000 in.3

V = 750 in.3

V = 500 in.3

Part C The volumes of the other two snowballs are 750 in. 3 and 1,000 in. 3 . How tall is the snowman?

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