Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 6 - Radical Functions and Rational Exponents - 6-2 Multiplying and Dividing Radical Expressions - Practice and Problem-Solving Exercises - Page 372: 78

Answer

$\dfrac{\sqrt[6]{x^4y^3}}{|y|}$

Work Step by Step

Using $a^{-m}=\dfrac{1}{a^m}$ and $\dfrac{1}{a^{-m}}=a^m,$ the given expression, $ \sqrt[6]{\dfrac{y^{-3}}{x^{-4}}} ,$ is equivalent to \begin{align*}\require{cancel} &\sqrt[6]{\dfrac{x^4}{y^3}} .\end{align*} Multiplying the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index, the expression above is equivalent to \begin{align*}\require{cancel} & \sqrt[6]{\dfrac{x^4}{y^3}\cdot\dfrac{y^3}{y^3}} \\\\&= \sqrt[6]{\dfrac{x^4y^3}{y^6}} .\end{align*} Using the properties of radicals, the expression above is equivalent to \begin{align*}\require{cancel} & \dfrac{\sqrt[6]{x^4y^3}}{\sqrt[6]{y^6}} \\\\&= \dfrac{\sqrt[6]{x^4y^3}}{\sqrt[6]{(y)^6}} \\\\&= \dfrac{\sqrt[6]{x^4y^3}}{|y|} &\left(\sqrt[n]{a^n}=|a|\text{ if $n$ is even} \right) .\end{align*} Hence, the simplified form of the given expression is $ \dfrac{\sqrt[6]{x^4y^3}}{|y|} .$
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