Practice and Problem-Solving Exercises

See Problem 1.

A Practice

Simplify each expression.

10. 36 1 2 36 super and 1 half end super
11. 27 1 3 27 super and 1 third end super
12. 49 1 2 49 super and 1 half end super
13. 10 1 2 ⋅ 10 1 2 10 super and 1 half end super , dot , 10 super and 1 half end super
14. ( − 3 ) 1 3 ⋅ ( − 3 ) 1 3 ⋅ ( − 3 ) 1 3 open , negative 3 , close super 1 third end super . dot . open , negative 3 , close super 1 third end super . dot . open , negative 3 , close super 1 third end super
15. 7 1 2 ⋅ 21 1 2 7 super and 1 half end super , dot , 21 super and 1 half end super
16. 2 1 2 ⋅ 32 1 2 2 super and 1 half end super , dot , 32 super and 1 half end super
17. 3 1 3 ⋅ 9 1 3 3 super and 1 third end super , dot , 9 super and 1 third end super
18. 3 1 4 ⋅ 27 1 4 3 super and 1 fourth end super , dot , 27 super and 1 fourth end super

See Problem 2.

Write each expression in radical form.

19. x 1 6 x super 1 sixth end super
20. x 1 5 x super 1 fifth end super
21. x 2 7 x super 2 sevenths end super
22. y 2 5 y super 2 fifths end super
23. y − 9 8 y super negative , 9 eighths end super
24. t − 3 4 t super negative , 3 fourths end super
25. x 1.5 x to the 1.5
26. y 1.2 y to the 1.2

Write each expression in exponential form.

27. − 10 square root of negative 10 end root
28. 7 x 3 square root of 7 , x cubed end root
29. ( 7 x ) 3 square root of open , 7 x , close cubed end root
30. ( 7 x ) 3 open , square root of 7 x end root , close cubed
31. a 2 3 cube root of eh squared end root ,
32. ( a 3 ) 2 open , cube root of eh , , close squared
33. c 2 4 the fourth , root of c squared end root ,
34. ( 5 x y ) 6 3 cube root of open , 5 x y , close to the sixth end root ,

See Problem 3.

Optimal Height The optimal height h of the letters of a message printed on pavement is given by the formula h = 0.00252 d 2.27 e . h equals . fraction 0.00252 . d to the 2.27 , over e end fraction . .  Here d is the distance of the driver from the letters and e is the height of the driver's eye above the pavement. All of the distances are in meters. Find h for the given values of d and e.

35. d = 100  m , e = 1.2  m d equals 100 , m comma e equals 1.2 , m
36. d = 50  m , e = 1.2  m d equals 50 m comma e equals 1.2 , m
37. d = 50  m , e = 2.3  m d equals 50 m comma e equals 2.3 , m
38. d = 25  m , e = 2.3  m d equals 25 m comma e equals 2.3 , m

See Problem 4.

Find each product or quotient.

39. ( 6 4 ) ( 6 3 ) open , the fourth , root of 6 , , close . open , cube root of 6 , , close
40. y 3 9 y 9 3 fraction the ninth , root of y cubed end root , , over cube root of y to the ninth end root , end fraction
41. 5 ⋅ 5 5 square root of 5 dot , the fifth , root of 5 ,
42. 7 7 ⋅ 7 3 the seventh , root of 7 , , dot , cube root of 7 ,
43. 4 6 4 3 fraction the sixth , root of 4 , , over cube root of 4 , end fraction
44. 18 4 ⋅ 12 the fourth , root of 18 , , dot square root of 12
45. 6 36 3 fraction square root of 6 , over cube root of 36 , end fraction
46. x 4 y x 2 y 8 4 fraction square root of x to the fourth , y end root , over the fourth , root of x squared , y to the eighth end root , end fraction

See Problem 5.

Simplify each number.

47. 8 2 3 8 super and 2 thirds end super
48. 64 2 3 64 2 3 64 super and 2 thirds end super . 64 super and 2 thirds end super
49. ( − 8 ) 2 3 open , negative 8 , close super 2 thirds end super
50. ( − 32 ) 6 5 open , negative 32 , close super 6 fifths end super
51. ( 32 ) − 4 5 open 32 close super negative , 4 fifths end super
52. 4 1.5 4 to the 1.5
53. 16 1.5 16 to the 1.5
54. 10,000 0.75 10,000 to the 0.75

See Problem 6.

Write each expression in simplest form.

55. ( x 2 3 ) − 3 open , x super 2 thirds end super , close super negative 3 end super
56. ( x − 4 7 ) 7 open . x super negative , 4 sevenths end super . close to the seventh
57. ( 3 x 2 3 ) − 1 open . 3 , x super 2 thirds end super . close super negative 1 end super
58. 5 ( x 2 3 ) − 1 5 . open , x super 2 thirds end super , close super negative 1 end super
59. ( − 27 x − 9 ) 1 3 open . negative 27 , x super negative 9 end super . close super 1 third end super
60. ( − 32 y 15 ) 1 5 open . negative 32 , y to the fifteenth . close super 1 fifth end super
61. ( x 1 2 y − 2 3 ) − 6 open . x super 1 half end super . y super negative , 2 thirds end super . close super negative 6 end super
62. ( x 2 3 y − 1 6 ) − 12 open . x super 2 thirds end super . y super negative , 1 sixth end super . close super negative 12 end super
63. ( x 3 x − 1 ) − 1 4 open . fraction x cubed , over x super negative 1 end super end fraction . close super negative , 1 fourth end super
64. ( x 2 x − 11 ) 1 3 open . fraction x squared , over x super negative 11 end super end fraction . close super 1 third end super
65. ( x 1 4 y − 3 4 ) 12 open . fraction x super 1 fourth end super , over y super negative , 3 fourths end super end fraction . close to the twelfth
66. ( x − 2 3 y − 1 3 ) 15 open . fraction x super negative , 2 thirds end super , over y super negative , 1 third end super end fraction . close to the fifteenth

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