Answer
$2\sqrt[4]{2}$
Work Step by Step
Using $\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}$ or the product rule of radicals, the given expression, $
\sqrt[4]{4}\cdot\sqrt[4]{8}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[4]{4(8)}
\\\\=
\sqrt[4]{32}
\\\\=
\sqrt[4]{16\cdot2}
\\\\=
\sqrt[4]{(2)^2\cdot2}
\\\\=
2\sqrt[4]{2}
.\end{array}
