54. Physics The formula F = m v 2 r f equals . fraction m , v squared , over r end fraction  gives the centripetal force F of an object of mass m moving along a circle of radius r, where v is the tangential velocity of the object. Solve the formula for v. Rationalize the denominator.
55. Satellites The circular velocity v in miles per hour of a satellite orbiting Earth is given by the formula v = 1.24 × 10 12 r , v equals . square root of fraction 1.24 , times , 10 to the twelfth , over r end fraction end root . comma  where r is the distance in miles from the satellite to the center of the Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (The radius of the Earth is 3950 mi.)
56. 1. Simplify 2 + 3 75 fraction square root of 2 plus square root of 3 , over square root of 75 end fraction  by multiplying the numerator and denominator by 75 . square root of 75 .
    2. Simplify the expression in (a) by multiplying by 3 square root of 3  instead of 75 . square root of 75 .
    3. Explain how you would simplify 2 + 3 98 . fraction square root of 2 plus square root of 3 , over square root of 98 end fraction . .

Simplify each expression. Rationalize all denominators.

57. 5 ⋅ 50 square root of 5 dot square root of 50
58. 4 3 ⋅ 80 3 cube root of 4 , , dot , cube root of 80 ,
59. x 5 y 5 ⋅ 3 2 x 7 y 6 square root of x to the fifth , y to the fifth end root . dot 3 . square root of 2 , x to the seventh , y to the sixth end root
60. 5 2 x y 6 ⋅ 2 2 x 3 y 5 . square root of 2 x , y to the sixth end root . dot 2 . square root of 2 , x cubed , y end root
61. 2 ( 50 + 7 ) square root of 2 . open , square root of 50 plus 7 , close
62. 5 ( 5 + 15 ) square root of 5 . open . square root of 5 plus square root of 15 . close
63. 5 x 4 2 x 2 y 3 fraction square root of 5 , x to the fourth end root , over square root of 2 , x squared , y cubed end root end fraction
64. 5 2 3 7 x fraction 5 square root of 2 , over 3 , square root of 7 x end root end fraction
65. 1 9 x 3 fraction 1 , over cube root of 9 x end root , end fraction
66. 10 5 x 2 3 fraction 10 , over cube root of 5 , x squared end root , end fraction
67. 14 3 7 x 2 y 3 fraction cube root of 14 , , over cube root of 7 , x squared , y end root , end fraction
68. 3 11 x 3 y − 2 12 x 4 y fraction 3 . square root of 11 , x cubed , y end root , over negative 2 . square root of 12 , x to the fourth , y end root end fraction
69. Physics The mass m of an object is 80 square root of 80  g and its volume V is 5  cm  3 . square root of 5 , cm cubed . . Use the formula D = m V d equals , m over v  to find the density D of the object.
70. Writing Does x 3 = x 2 3 square root of x cubed end root , equals , cube root of x squared end root ,  for all, some, or no values of x? Explain.
71. Open-Ended Of the equivalent expressions 2 3 , 2 3 square root of 2 thirds end root , comma , fraction square root of 2 , over square root of 3 end fraction  and 6 3 , fraction square root of 6 , over 3 end fraction , comma  which do you prefer to use for finding a decimal approximation with a calculator? Justify your reasoning.

72. Error Analysis Explain the error in this simplification of radical expressions.

    

Determine whether each expression is always, sometimes, or never a real number. Assume that x can be any real number.

73. − x 2 3 cube root of negative , x squared end root ,
74. − x 2 square root of negative , x squared end root
75. − x square root of negative x end root

C Challenge

Simplify each expression. Rationalize all denominators.

76. 16 x 4 y 4 square root of square root of 16 , x to the fourth , y to the fourth end root end root
77. 8000 3 square root of cube root of 8000 , end root
78. y − 3 x − 4 6 the sixth , root of fraction y super negative 3 end super , over x super negative 4 end super end fraction end root ,
79. Reasoning When x a y b square root of x to the eh , y to the b end root  simplified, the result is 1 x c y 3 d , fraction 1 , over x to the c . y super 3 d end super end fraction . comma  where c and d are positive integers. Express a in terms of c, and b in terms of d.

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