B Apply

67. Think About a Plan The ratio R of radioactive carbon to nonradioactive carbon left in a sample of an organism that died T years ago can be approximated by the formula R = A ( 2.7 ) − T 8033 . r equals eh . open 2.7 close super negative , t over 8033 end super . .  Here A is the ratio of radioactive carbon to nonradioactive carbon in the living organism. What percent of A is left after 2000 years? After 4000 years? After 8000 years?
    • What are the known and unknown values?
    • How can you use the properties of exponents to solve this problem?
68. The expression 0.036 m 3 4 0.036 . m super 3 fourths end super  is used in the study of fluids. Which best represents the value of the expression for m = 46 × 10 4 ? m equals 46 times , 10 to the fourth , question mark
    1. 636
    2. 1460
    3. 1660
    4. 16,600

Simplify each number.

69. ( − 343 ) 1 3 open , negative 343 , close super 1 third end super
70. ( − 243 ) 1 5 open , negative 243 , close super 1 fifth end super
71. 32 1.2 32 to the 1.2
72. 243 1.2 243 to the 1.2
73. 64 3.5 64 to the 3.5
74. 100 4.5 100 to the 4.5
75. − ( − 27 ) − 4 3 negative . open , negative 27 , close super negative , 4 thirds end super
76. 1000 4 3 100 3 2 fraction 1000 super and 4 thirds end super , over 100 super and 3 halves end super end fraction
77. 25 3 2 25 super and 3 halves end super
78. Science A desktop world globe has a volume of about 1386 cubic inches. The radius of Earth is approximately equal to the radius of the globe raised to the 10th power. Find the radius of Earth. (Hint: Use the formula V = 4 3 π r 3 v equals , 4 thirds , pi , r cubed  for the volume of a sphere.)

Simplify each expression.

79. x 2 7 ⋅ x 3 14 x super 2 sevenths end super , dot , x super 3 fourteenths end super
80. y 1 2 ⋅ y 3 10 y super 1 half end super , dot , y super 3 tenths end super
81. x 3 5 ÷ x 3 10 x super 3 fifths end super , divides , x super 3 tenths end super
82. y 5 7 ÷ y 3 14 y super 5 sevenths end super , divides , y super 3 fourteenths end super
83. x 2 3 y − 1 4 x 1 2 y − 1 2 fraction x super 2 thirds end super . y super negative , 1 fourth end super , over x super 1 half end super . y super negative , 1 half end super end fraction
84. x 1 2 y − 1 3 x 3 4 y 1 2 fraction x super 1 half end super . y super negative , 1 third end super , over x super 3 fourths end super . y super 1 half end super end fraction
85. ( 16 x 14 81 y 18 ) 1 2 open . fraction 16 , x to the fourteenth , over 81 , y to the eighteenth end fraction . close super 1 half end super
86. ( 81 y 16 16 x 12 ) 1 2 open . fraction 81 , y to the sixteenth , over 16 , x to the twelfth end fraction . close super 1 half end super
87. ( 8 x 6 27 y 9 ) 1 3 open . fraction 8 , x to the sixth , over 27 , y to the ninth end fraction . close super 1 third end super
88. Open-Ended Find three nonzero numbers a such that a ( 4 + 5 1 2 ) eh . open . 4 plus , 5 super and 1 half end super . close  is a rational number. Can a itself be a rational number? Explain.
89. 1. Reasoning Show that x 2 4 = x the fourth , root of x squared end root , , equals square root of x  by using the definition of fourth root.
    2. Reasoning Show that x 2 4 = x the fourth , root of x squared end root , , equals square root of x  by rewriting x 2 4 the fourth , root of x squared end root ,  in exponential form.

90. Simplify 4 1 2 ⋅ 4 1 2 4 super and 1 half end super , dot , 4 super and 1 half end super  using the following methods. Show all your work.

    1. Use the properties of exponents.
    2. Simplify each term in the product, then multiply.
    3. Convert to radical form, then simplify.

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