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: 62

Answer

$\color{blue}{\dfrac{1000}{1331}}$

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, $\sqrt[n]{a^n} = n$ (6) $\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$ Use rule (4) above to obtain: $\left(\dfrac{121}{100}\right)^{-3/2} = \dfrac{1}{(\frac{121}{100})^{3/2}}$ Use rule (1) above to obtain: $=\dfrac{1}{[(\frac{121}{100})^{1/2}]^3}$ Use rule (2) above to obtain: $=\dfrac{1}{\left(\sqrt[2]{\frac{121}{100}}\right)^3}$ With $\dfrac{121}{100}=\left(\dfrac{11}{10}\right)^2$, the expression above is equivalent to: $=\dfrac{1}{\left(\sqrt[2]{(\frac{11}{10})^2}\right)^3}$ Use rule (3) above to obtain: $=\dfrac{1}{\left(\frac{11}{10}\right)^3}$ Use rule (6) above to obtain: $\\=\dfrac{1}{\frac{11^3}{10^3}} \\=\dfrac{1}{\frac{1331}{1000}} \\=1 \cdot \dfrac{1000}{1331} \\=\color{blue}{\dfrac{1000}{1331}}$
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