Practice and Problem-Solving Exercises

See Problem 1.

A Practice

Multiply, if possible. Then simplify.

10. 8 ⋅ 32 square root of 8 dot square root of 32
11. 4 3 ⋅ 16 3 cube root of 4 , , dot , cube root of 16 ,
12. 9 3 ⋅ − 81 3 cube root of 9 , , dot , cube root of negative 81 end root ,
13. 8 4 ⋅ 32 3 the fourth , root of 8 , , dot , cube root of 32 ,
14. − 5 ⋅ 5 square root of negative 5 end root , dot square root of 5
15. − 5 3 ⋅ − 25 3 cube root of negative 5 end root , , dot , cube root of negative 25 end root ,
16. 9 3 ⋅ − 24 3 cube root of 9 , , dot , cube root of negative 24 end root ,
17. − 12 3 ⋅ − 18 3 cube root of negative 12 end root , , dot , cube root of negative 18 end root ,
18. 50 ⋅ 75 square root of 50 dot square root of 75

See Problem 2.

Simplify.

19. 20 x 3 square root of 20 , x cubed end root
20. 81 x 3 3 cube root of 81 , x cubed end root ,
21. 50 x 5 square root of 50 , x to the fifth end root
22. 32 a 5 3 cube root of 32 , eh to the fifth end root ,
23. 54 y 10 3 cube root of 54 , y to the tenth end root ,
24. 200 a 6 b 7 square root of 200 , eh to the sixth , b to the seventh end root
25. − 250 x 6 y 5 3 cube root of negative 250 , x to the sixth , y to the fifth end root ,
26. 64 x 3 y 6 4 the fourth , root of 64 , x cubed , y to the sixth end root ,
27. − 32 x 6 x 7 5 the fifth , root of negative 32 , x to the sixth , x to the seventh end root ,

See Problem 3.

Multiply and simplify.

28. 6 3 ⋅ 16 3 cube root of 6 , , dot , cube root of 16 ,
29. 8 y 5 ⋅ 40 y 2 square root of 8 , y to the fifth end root , dot . square root of 40 , y squared end root
30. 8 x 5 ⋅ 3 x square root of 8 , x to the fifth end root , dot , square root of 3 x end root
31. 4 2 x ⋅ 5 6 x y 2 4 , square root of 2 x end root , dot 5 . square root of 6 x , y squared end root
32. 3 5 y 3 3 ⋅ 2 50 y 4 3 3 . cube root of 5 , y cubed end root , . dot 2 . cube root of 50 , y to the fourth end root ,
33. − 2 x 2 y 2 3 ⋅ 2 15 x 5 y 3 negative . cube root of 2 , x squared , y squared end root , . dot 2 . cube root of 15 , x to the fifth , y end root ,
34. 81 x 5 y 4 4 ⋅ 32 x 3 y 4 the fourth , root of 81 , x to the fifth , y to the fourth end root , . dot . the fourth , root of 32 , x cubed , y end root ,
35. 2 2 x y 2 3 ⋅ 4 x 2 y 5 3 2 . cube root of 2 x , y squared end root , . dot . cube root of 4 , x squared , y to the fifth end root ,
36. 3 18 a 9 4 ⋅ 6 a b 2 4 3 . the fourth , root of 18 , eh to the ninth end root , . dot . the fourth , root of 6 eh , b squared end root ,

See Problem 4.

Divide and simplify.

37. 500 5 fraction square root of 500 , over square root of 5 end fraction
38. 48 x 3 3 x y 2 fraction square root of 48 , x cubed end root , over square root of 3 x , y squared end root end fraction
39. 56 x 5 y 5 7 x y fraction square root of 56 , x to the fifth , y to the fifth end root , over square root of 7 x y end root end fraction
40. 250 x 7 y 3 3 2 x 2 y 3 fraction cube root of 250 , x to the seventh , y cubed end root , , over cube root of 2 , x squared , y end root , end fraction
41. 48 x 3 y 2 3 6 x 4 y 3 fraction cube root of 48 , x cubed , y squared end root , , over cube root of 6 , x to the fourth , y end root , end fraction
42. 20 a b 45 a 2 b 3 fraction square root of 20 eh b end root , over square root of 45 , eh squared , b cubed end root end fraction

See Problem 5.

Rationalize the denominator of each expression.

43. x 2 fraction square root of x , over square root of 2 end fraction
44. 5 8 x fraction square root of 5 , over square root of 8 x end root end fraction
45. x 3 2 3 fraction cube root of x , , over cube root of 2 , end fraction
46. 5 3 x 3 cube root of fraction 5 , over 3 x end fraction end root ,
47. 2 4 5 4 fraction the fourth , root of 2 , , over the fourth , root of 5 , end fraction
48. 15 60 x 5 3 12 x fraction 15 . square root of 60 , x to the fifth end root , over 3 , square root of 12 x end root end fraction
49. 3 x y 2 5 x y 3 fraction square root of 3 x , y squared end root , over square root of 5 x , y cubed end root end fraction
50. 5 x 4 y 2 x 2 y 3 fraction square root of 5 , x to the fourth , y end root , over square root of 2 , x squared , y cubed end root end fraction
51. 12 a b 3 c 2 3 10 a 3 b c 3 fraction cube root of 12 eh , b cubed , c squared end root , , over cube root of 10 , eh cubed , b c end root , end fraction

    B Apply

52. Think About a Plan The formula t = 2 s a t equals , square root of fraction 2 s , over eh end fraction end root  shows the time t that any vehicle takes to travel a distance s at a constant acceleration a, starting from rest. What is the difference in time between a car accelerating at 16  m/s 2 16 , mslashs squared and one accelerating at 25  m/s 2 25 , mslashs squared for a distance of 200 m?
    • What is the time that a car accelerating at 16  m/s 2 16 , mslashs squared takes to travel 200 m?
    • What is the time that a car accelerating at 25  m/s 2 25 , mslashs squared takes to travel 200 m?
53. Geometry The base of a triangle is 18 square root of 18  cm and its height is 8 square root of 8  cm. Find its area.

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