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 - 10.2 Rational Numbers as Exponents - 10.2 Exercise Set: 96

Answer

$\sqrt[12]{n}$

Work Step by Step

Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then \begin{array}{l}\require{cancel} \sqrt[6]{\sqrt{n}} \\\\= \sqrt[6]{n^{1/2}} \\\\= \left( n^{1/2} \right)^{1/6} .\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} n^{\frac{1}{2}\cdot\frac{1}{6}} \\\\= n^{\frac{1}{12}} .\end{array} Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then \begin{array}{l}\require{cancel} n^{\frac{1}{12}} \\\\= \sqrt[12]{n^1} \\\\= \sqrt[12]{n} .\end{array}
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