SECTION A.10 nth Roots; Rational Exponents A95 In Problems 119–128, expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. 119. x x x x 1 3 2 1 1 3 2 1 2 ( ) ( ) + + ⋅ + ≥− 120. x x x x 4 4 3 4 2 2 4 3 2 1 3 ( ) ( ) + + ⋅ + ⋅ 121. x x x x x x 6 8 8 0 1 2 2 3 2 1 2 ( ) + − − ≥ 122. x x x x 6 2 3 8 0 1 2 3 2 ( ) + + ⋅ ≥ 123. x x x x 3 4 4 4 2 2 4 3 2 1 3 ( ) ( ) + + ⋅ + ⋅ 124. x x x x 2 3 4 4 3 4 4 3 2 1 3 ( ) ( ) + + ⋅ + 125. x x x x x 4 3 5 2 3 3 3 5 2 3 3 2 1 3 3 2 4 3 1 2 ( ) ( ) ( ) ( ) + + + + + ≥− 126. x x x x x 6 6 1 4 3 6 6 1 4 3 3 4 1 3 3 2 4 3 1 2 ( ) ( ) ( ) ( ) + − + + − ≥ 127. x x x 3 3 2 0 1 2 1 2 + > − 128. x x x 8 4 0 1 3 2 3 − ≠ − In Problems 129–136, use a calculator to approximate each radical. Round your answer to two decimal places. 129. 2 130. 7 131. 4 3 132. 5 3 − 133. 2 3 3 5 + − 134. 5 2 2 4 − + 135. 3 5 2 3 3 − 136. 2 3 4 2 3 − 137. Calculating the Amount of Gasoline in a Tank A station stores its gasoline in underground tanks that are right circular cylinders lying on their sides. See the figure. The volume V of gasoline in the tank (in gallons) is given by the formula V h h 40 96 0.608 2 = − where h is the height of the gasoline (in inches) as measured on a depth stick. (a) If h 12 = inches, how many gallons of gasoline are in the tank? (b) If h 1 = inch, how many gallons of gasoline are in the tank? 138. Inclined Planes The final velocity v of an object in feet per second (ft/sec) after it slides down a frictionless inclined plane of height h feet is v h v 64 0 2 = + where v0 is the initial velocity (in ft/sec) of the object. (a) What is the final velocity v of an object that slides down a frictionless inclined plane of height 4 feet? Assume that the initial velocity is 0. (b) What is the final velocity v of an object that slides down a frictionless inclined plane of height 16 feet? Assume that the initial velocity is 0. (c) What is the final velocity v of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/s? ‘Are You Prepared?’ Answers 1. 9; 9− 2. 4; 4 h v v0
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