Chapter 10 - Exponents and Radicals - 10.2 Rational Numbers as Exponents - 10.2 Exercise Set - Page 641: 101

$\sqrt[12]{xy}$

Using $a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$, the given expression, $ \sqrt[3]{\sqrt[4]{xy}} ,$ is equivalent to \begin{array}{l}\require{cancel} \left( \sqrt[4]{xy} \right)^{\frac{1}{3}} \\\\= \left( xy^{\frac{1}{4}} \right)^{\frac{1}{3}} \\\\= \left( xy \right)^{\frac{1}{4}\cdot\frac{1}{3}} \\\\= \left( xy \right)^{\frac{1}{12}} \\\\= \sqrt[12]{(xy)^1} \\\\= \sqrt[12]{xy} .\end{array}