Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 10 - Exponents and Radicals - Review Exercises: Chapter 10: 38

Answer

$\sqrt[12]{x^5}$

Work Step by Step

Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[3]{x^2}}{\sqrt[4]{x}} \\\\= \dfrac{x^{2/3}}{x^{1/4}} .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{x^{2/3}}{x^{1/4}} \\\\= x^{\frac{2}{3}-\frac{1}{4}} \\\\= x^{\frac{8}{12}-\frac{3}{12}} \\\\= x^{\frac{5}{12}} .\end{array} Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the given expression is equivalent to \begin{array}{l}\require{cancel} x^{\frac{5}{12}} \\\\= \sqrt[12]{x^5} .\end{array}
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