Answer
$4\sqrt[3]{5}$
Work Step by Step
Extracting the factors that are perfect powers of the index, the given expression, $
\sqrt[3]{4}\cdot\sqrt[3]{80}
,$ is equivalent to
\begin{align*}\require{cancel}
&
=\sqrt[3]{4}\cdot\sqrt[3]{8\cdot10}
\\\\&=
\sqrt[3]{4}\cdot\sqrt[3]{(2)^3\cdot10}
\\\\&=
\sqrt[3]{4}\cdot2\sqrt[3]{10}
.\end{align*}
Using $\sqrt[n]{x}\cdot\sqrt[n]{y}=\sqrt{xy},$ the expression above is equivalent to
\begin{align*}
&
=2\sqrt[3]{4(10)}
\\\\&=
2\sqrt[3]{40}
.\end{align*}
Extracting the factors that are perfect powers of the index, the expression above is equivalent to
\begin{align*}\require{cancel}
&
=2\sqrt[3]{8\cdot5}
\\\\&=
2\sqrt[3]{(2)^3\cdot5}
\\\\&=
2(2)\sqrt[3]{5}
\\\\&=
4\sqrt[3]{5}
.\end{align*}
Hence, the simplified form of the given expression is $
4\sqrt[3]{5}
$.
