Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter R - Review of Basic Concepts - R.6 Rational Exponents - R.6 Exercises: 60

Answer

$\color{blue}{\dfrac{1}{16}}$

Work Step by Step

RECALL: (1) $a^{m/n} = \left(a^{1/n}\right)^m$ (2) $a^{1/n} = \sqrt[n]{a}$ (3) For positive real numbers $a$, $\sqrt[n]{a^n}=a$ (4) $a^{-m} = \dfrac{1}{a^m}$ (5) When $n$ is odd, $\sqrt[n]{a^n} = n$ Use rule (4) above to obtain: $(-32)^{-4/5} = \dfrac{1}{(-32)^{4/5}}$ Use rule (1) above to obtain: $=\dfrac{1}{[(-32)^{1/5}]^4}$ Use rule (2) above to obtain: $=\dfrac{1}{(\sqrt[5]{-32})^4}$ With $-32=(-2)^5$, the expression above is equivalent to: $=\dfrac{1}{\left(\sqrt[5]{(-2)^5}\right)^4}$ Use rule (5) above to obtain: $=\dfrac{1}{(-2)^4} \\=\color{blue}{\dfrac{1}{16}}$
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