Answer
$6x^2|y|\sqrt[4]{2y}$
Work Step by Step
Extracting the perfect powers of the index, the given expression, $ \sqrt[4]{81x^5y^4}\cdot\sqrt[4]{32x^3y} ,$ is equivalent to
\begin{align*}
&
\sqrt[4]{81x^4y^4\cdot x}\cdot\sqrt[4]{16\cdot2x^3y}
\\\\&=
\sqrt[4]{(3xy)^4\cdot x}\cdot\sqrt[4]{(2)^4\cdot2x^3y}
\\\\&=
|3xy|\sqrt[4]{x}\cdot|2|\sqrt[4]{2x^3y}
\\\\&=
3|xy|\sqrt[4]{x}\cdot2\sqrt[4]{2x^3y}
.\end{align*}
Using $a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy},$ the expression above is equivalent to
\begin{align*}
& 3(2)|xy|\sqrt[4]{x(2x^3y)}
\\\\&=
6|xy|\sqrt[4]{2x^4y}
.\end{align*}
Extracting the perfect powers of the index, the expression above is equivalent to
\begin{align*} &
6|xy|\sqrt[4]{x^4\cdot2y}
\\\\&=
6|xy|\sqrt[4]{(x)^4\cdot2y}
\\\\&=
6|xy|\cdot|x|\sqrt[4]{2y}
\\\\&=
6|x^2y|\sqrt[4]{2y}
\\\\&=
6x^2|y|\sqrt[4]{2y}
.\end{align*}
Hence, the simplified form of the given expression is $
6x^2|y|\sqrt[4]{2y}
$.
