# Chapter 10 - Exponents and Radicals - 10.2 Rational Numbers as Exponents - 10.2 Exercise Set - Page 641: 96

$\sqrt[12]{n}$

#### 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}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.