Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises: 101

Answer

$x^{1/24}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $ \sqrt{\sqrt[3]{\sqrt[4]{x}}} .$ $\bf{\text{Solution Details:}}$ 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 expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[2]{\sqrt[3]{x^{1/4}}} \\\\= \sqrt[2]{(x^{1/4})^{1/3}} \\\\= ((x^{1/4})^{1/3})^{1/2} .\end{array} Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to \begin{array}{l}\require{cancel} x^{\frac{1}{4}\cdot\frac{1}{3}\cdot\frac{1}{2}} \\\\= x^{\frac{1}{24}} \\\\= x^{1/24} .\end{array}
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