Chapter 10 - Exponents and Radicals - 10.2 Rational Numbers as Exponents - 10.2 Exercise Set - Page 641: 102

$\sqrt[15]{2a}$

Using $a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$, the given expression, $ \sqrt[5]{\sqrt[3]{2a}} ,$ is equivalent to \begin{array}{l}\require{cancel} \left( \sqrt[3]{2a} \right)^{\frac{1}{5}} \\\\= \left( (2a)^{\frac{1}{3}} \right)^{\frac{1}{5}} \\\\= \left( 2a \right)^{\frac{1}{3}\cdot\frac{1}{5}} \\\\= \left( 2a \right)^{\frac{1}{15}} \\\\= \sqrt[15]{(2a)^1} \\\\= \sqrt[15]{2a} .\end{array}