Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises: 34

Answer

$\frac{1}{729}$

Work Step by Step

We know that $a^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}$, where $a^{\frac{m}{n}}$ is a real number. Therefore, $81^{-\frac{3}{2}}=\frac{1}{81^{\frac{3}{2}}}$. We know that $a^{\frac{m}{n}}=\sqrt[n] a^{m}=\sqrt[n] (a)^{m}$, where all indicated roots are real numbers. Therefore, $\frac{1}{81^{\frac{3}{2}}}=\frac{1}{\sqrt 81^{3}}=\frac{1}{(\sqrt 81)^{3}}=\frac{1}{9^{3}}=\frac{1}{729}$. $\sqrt 81=9$, because $9^{2}=81$
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