A94 APPENDIX Review In Problems 69–76, rationalize the numerator of each expression. Assume that all variables are positive when they appear. 69. 11 1 2 + 70. 5 43 3 − 71. 6 15 15 − 72. 5 3 5 + 73. x c x c x c − − ≠ 74. x x x 2 4 4 − − ≠ 75. x x x 7 1 8 8 − − − ≠ 76. x x x 4 9 25 25 − − − ≠ In Problems 77–80, solve each equation. 77. t2 1 2 3 − = 78. t3 1 2 3 + = − 79. x x 15 2 − = 80. x x 12 − = In Problems 81–96, simplify each expression. 81. 82 3 82. 43 2 83. 64 1 3 ( ) − 84. 163 4 85. 1003 2 86. 253 2 87. 4 3 2 − 88. 16 3 2 − 89. 9 8 3 2 ( ) 90. 27 8 2 3 ( ) 91. 8 9 3 2 ( )− 92. 8 27 2 3 ( )− 93. 1000 1 3 ( ) − − 94. 25 1 2 − − 95. 64 125 2 3 ( ) − − 96. 81 3 4 − − In Problems 97–104, simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. 97. x x x 3 4 1 3 1 2 − 98. x x x 2 3 1 2 1 4 − 99. x y3 6 1 3 ( ) 100. x y4 8 3 4 ( ) 101. x y xy x y 2 1 3 2 2 3 2 3 2 3 ( ) ( ) 102. xy x y x y 1 4 2 2 1 2 2 3 4 ( ) ( ) ( ) 103. x y xy 16 2 1 3 3 4 2 1 4 ( ) ( ) − 104. x y xy 4 1 1 3 3 2 3 2 ( ) ( ) − Applications and Extensions In Problems 105–118, expressions that occur in calculus are given.Write each expression as a single quotient in which only positive exponents and/or radicals appear. 105. x x x x 1 2 1 1 1 2 1 2 ( ) ( ) + + + >− 106. x x x x 1 2 0 1 2 1 2 + + > 107. x x x x x 2 1 1 2 1 2 2 1 2 2 2 1 2 ( ) ( ) + + ⋅ + ⋅ − 108. x x x x 1 1 3 1 1 1 3 2 3 ( ) ( ) + + ⋅ + ≠ − − 109. x x x x x 4 3 1 2 5 5 1 5 4 3 5 + ⋅ − + − ⋅ + > 110. x x x x x x 8 1 3 2 2 24 8 1 2, 1 8 3 2 3 3 2 3 ( ) ( ) + − + − + ≠ ≠ − 111. x x x x x 1 1 2 1 1 1 + − ⋅ + + >− 112. x x x x x 1 2 2 1 1 2 2 2 + − ⋅ + + 113. x x x x x 4 2 4 4 4 1 2 1 2 ( ) ( ) + − + + >− − 114. x x x x x 9 9 9 3 3 2 1 2 2 2 1 2 2 ( ) ( ) − + − − − < < − 115. x x x x x x 1 1 1 or 1 2 2 1 2 2 1 2 2 ( ) ( ) − − − <− > 116. x x x x 4 4 4 2 1 2 2 2 1 2 2 ( ) ( ) + − + + − 117. x x x x x x 1 2 2 1 0 2 2 2 ( ) + − + > 118. x x x x x x x 2 1 2 3 1 1 1, 1 2 1 3 3 2 2 3 2 2 3 ( ) ( ) ( ) − + − − ≠ − ≠ −
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