Intermediate Algebra (12th Edition)

Published by Pearson

Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises: 102

Answer

$y^{1/30}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\sqrt[3]{\sqrt[5]{\sqrt[]{y}}} .$ $\bf{\text{Solution Details:}}$ 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[3]{\sqrt[5]{\sqrt[2]{y}}} \\\\= \sqrt[3]{\sqrt[5]{y^{1/2}}} \\\\= \sqrt[3]{(y^{1/2})^{1/5}} \\\\= ((y^{1/2})^{1/5})^{1/3} .\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} y^{\frac{1}{2}\cdot\frac{1}{5}\cdot\frac{1}{3}} \\\\= y^{\frac{1}{30}} \\\\= y^{1/30} .\end{array}

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