College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter P, Prerequisites - Section P.4 - Rational Exponents and Radicals - P.4 Exercises: 81

Answer

$\dfrac{x^{\frac{1}{4}}y^{\frac{1}{4}}}{2}$

Work Step by Step

RECALL: (i) $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ (ii) $\dfrac{a^m}{a^n}=a^{m-n}, a\ne0$ (iii) $(am)^m = a^mb^m$ Simplify the denominator: $=\dfrac{\sqrt{xy}}{\sqrt[4]{2^4xy}} \\=\dfrac{\sqrt{xy}}{2\sqrt[4]{xy}} \\=\dfrac{1}{2} \cdot \dfrac{\sqrt{xy}}{\sqrt[4]{xy}}$ Use rule (i) above to obtain: $=\dfrac{1}{2} \cdot \dfrac{(xy)^{\frac{1}{2}}}{(xy)^{\frac{1}{4}}}$ Use rule (ii) above to obtain: $=\frac{1}{2} \cdot (xy)^{\frac{1}{2} - \frac{1}{4}} \\=\frac{1}{2} \cdot (xy)^{\frac{2}{4} - \frac{1}{4}} \\=\frac{1}{2} \cdot (xy)^{\frac{1}{4}} \\=\dfrac{(xy)^{\frac{1}{4}}}{2}$ Use rule (iii) above to obtain: $\\=\dfrac{x^{\frac{1}{4}}y^{\frac{1}{4}}}{2}$
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