College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter R - Section R.6 - Rational Exponents - R.6 Exercises - Page 58: 115

Answer

$\dfrac{1}{100}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $ \dfrac{2^{2/3}}{2000^{2/3}} ,$ use the laws of exponents. $\bf{\text{Solution Details:}}$ Using the extended Power Rule of the laws of exponents which states that $\left( \dfrac{x^m}{z^p} \right)^q=\dfrac{x^{mq}}{z^{pq}},$ the expression above is equivalent to\begin{array}{l}\require{cancel} \left(\dfrac{2}{2000}\right)^{2/3} \\\\= \left(\dfrac{1}{1000}\right)^{2/3} .\end{array} Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \left(\sqrt[3]{\dfrac{1}{1000}}\right)^{2} \\\\= \left(\sqrt[3]{\left(\dfrac{1}{10}\right)^3}\right)^{2} \\\\= \left(\dfrac{1}{10}\right)^{2} \\\\= \dfrac{1}{100} .\end{array}
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