Answer
$4xy^2\sqrt[3]{y}$
Work Step by Step
Extracting the perfect powers of the index, the given expression, $
2\sqrt[3]{2xy^2}\cdot\sqrt[3]{4x^2y^5}
,$ is equivalent to
\begin{align*}
&
=2\sqrt[3]{2xy^2}\cdot\sqrt[3]{y^3\cdot4x^2y^2}
\\\\&=
2\sqrt[3]{2xy^2}\cdot\sqrt[3]{(y)^3\cdot4x^2y^2}
\\\\&=
2\sqrt[3]{2xy^2}\cdot y\sqrt[3]{4x^2y^2}
.\end{align*}
Using $a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy},$ the expression above is equivalent to
\begin{align*}
&
=2(y)\sqrt[3]{2xy^2(4x^2y^2)}
\\\\&=
2y\sqrt[3]{8x^3y^4}
.\end{align*}
Extracting the perfect powers of the index, the expression above is equivalent to
\begin{align*}
&
=2y\sqrt[3]{8x^3y^3\cdot y}
\\\\&=
2y\sqrt[3]{(2xy)^3\cdot y}
\\\\&=
2y(2xy)\sqrt[3]{y}
\\\\&=
4xy^2\sqrt[3]{y}
.\end{align*}
Hence, the simplified form of the given expression is $
4xy^2\sqrt[3]{y}
$.
