Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.2 - Exponents and Radicals - 1.2 Exercises - Page 23: 70

Answer

$a)$ $\dfrac{x}{ab^{10}y^{10/3}}$ $b)$ $4s^{3/2}t^{9/2}$

Work Step by Step

$a)$ $\Big(\dfrac{a^{1/6}b^{-3}}{x^{-1}y}\Big)^{3}\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)$ First, evaluate the power: $\Big(\dfrac{a^{1/6}b^{-3}}{x^{-1}y}\Big)^{3}\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)=\Big(\dfrac{a^{1/2}b^{-9}}{x^{-3}y^{3}}\Big)\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)=...$ Evaluate the product: $...=\dfrac{a^{1/2}b^{-10}x^{-2}}{a^{3/2}x^{-3}y^{10/3}}=...$ Finally, evaluate the division and simplify if possible: $...=\dfrac{a^{1/2-3/2}b^{-10}x^{-2+3}}{y^{10/3}}=\dfrac{a^{-1}b^{-10}x}{y^{10/3}}=\dfrac{x}{ab^{10}y^{10/3}}$ $b)$ $\dfrac{(9st)^{3/2}}{(27s^{3}t^{-4})^{2/3}}\Big(\dfrac{3s^{-2}}{4t^{1/3}}\Big)^{-1}$ Evaluate the powers in the numerator and the denominator of the first fraction and invert the second fraction to change the sign of its exponent: $\dfrac{(9st)^{3/2}}{(27s^{3}t^{-4})^{2/3}}\Big(\dfrac{3s^{-2}}{4t^{1/3}}\Big)^{-1}=\Big[\dfrac{(\sqrt{9^{3}})s^{3/2}t^{3/2}}{(\sqrt[3]{27^{2}})s^{2}t^{-8/3}}\Big]\Big(\dfrac{4t^{1/3}}{3s^{-2}}\Big)=...$ $...=\Big(\dfrac{27s^{3/2}t^{3/2}}{9s^{2}t^{-8/3}}\Big)\Big(\dfrac{4t^{1/3}}{3s^{-2}}\Big)=...$ Evaluate the product of fractions: $...=\dfrac{108s^{3/2}t^{3/2+1/3}}{27s^{2-2}t^{-8/3}}=\dfrac{108s^{3/2}t^{11/6}}{27t^{-8/3}}=...$ Evaluate the division and simplify: $...=\Big(\dfrac{108}{27}\Big)s^{3/2}t^{11/6+8/3}=4s^{3/2}t^{9/2}$
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