Answer
$x\sqrt[6]{xy^{5}} - \sqrt[15]{x^{13}y^{14}}$
Work Step by Step
Using $a(b+c)=ab+ac$, or the Distributive Property, the given expression, $
\sqrt[3]{x^2y} \left( \sqrt{xy}-\sqrt[5]{xy^3} \right)
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{x^2y} \left( \sqrt{xy} \right) - \sqrt[3]{x^2y} \left( \sqrt[5]{xy^3} \right)
\end{array}
Using the same indices for the radicals, the expression, $
\sqrt[3]{x^2y} \left( \sqrt{xy} \right) - \sqrt[3]{x^2y} \left( \sqrt[5]{xy^3} \right)$, simplifies to
\begin{array}{l}\require{cancel}
\sqrt[3(2)]{x^{2(2)}y^{1(2)}} \left( \sqrt[2(3)]{x^{1(3)}y^{1(3)}} \right) - \sqrt[3(5)]{x^{2(5)}y^{1(5)}} \left( \sqrt[5(3)]{x^{1(3)}y^{3(3)}} \right)
\\\\=
\sqrt[6]{x^{4}y^{2}} \left( \sqrt[6]{x^{3}y^{3}} \right) - \sqrt[15]{x^{10}y^{5}} \left( \sqrt[15]{x^{3}y^{9}} \right)
\\\\=
\sqrt[6]{x^{4}y^{2}(x^{3}y^{3})} - \sqrt[15]{x^{10}y^{5}(x^{3}y^{9})}
\\\\=
\sqrt[6]{x^{4+3}y^{2+3}} - \sqrt[15]{x^{10+3}y^{5+9}}
\\\\=
\sqrt[6]{x^{7}y^{5}} - \sqrt[15]{x^{13}y^{14}}
\\\\=
\sqrt[6]{x^{6}\cdot xy^{5}} - \sqrt[15]{x^{13}y^{14}}
\\\\=
x\sqrt[6]{xy^{5}} - \sqrt[15]{x^{13}y^{14}}
\end{array}
* Note that it is assumed that all variables represent positive numbers.
