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 - Page 64: 78



Work Step by Step

RECALL: (1) $a^m \cdot a^n = a^{m+n}$ (2) $\dfrac{a^m}{a^n} = a^{m-n}$ (3) $a^{1/n} = \sqrt[n]{a}$ (4) When $n$ is odd, $\sqrt[n]{a^n}=a$ (5) $a^{m/n} = \left(\sqrt[n]{a}\right)^m$ (6) $(ab)^m=a^mb^m$ (7) $(a^m)^n=a^{mn}$ (8) $a^{-m}=\dfrac{1}{a^m}$ Use rule (1) above to obtain: $=\dfrac{z^{1/3-2/3+1/6}}{(z^{-1/6})^3} \\=\dfrac{z^{2/6-4/6+1/6}}{(z^{-1/6})^{3}} \\=\dfrac{z^{-1/6}}{(z^{-1/6})^{3}}$ Use rule (7) above to obtain: $\\=\dfrac{z^{-1/6}}{z^{(-1/6) \cdot 3}} \\=\dfrac{z^{-1/6}}{z^{-3/6}}$ Use rule (2) above to obtain: $=z^{-1/6-(-3/6)} \\=z^{-1/6+3/6} \\=z^{2/6} \\=\color{blue}{z^{1/3}}$
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