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

$\sqrt[12]{a}$
$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\sqrt[6]{\sqrt{a}} .$ $\bf{\text{Solution Details:}}$ 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[6]{\sqrt{a}} \\\\= \sqrt[6]{a^{1/2}} \\\\= \left( a^{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} \left( a^{1/2} \right)^{1/6} \\\\= a^{\frac{1}{2}\cdot\frac{1}{6}} \\\\= a^{\frac{1}{12}} \\\\= \sqrt[12]{a} .\end{array}