Answer
$xy\sqrt[10]{x^7y^3}$
Work Step by Step
Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then
\begin{array}{l}\require{cancel}
\sqrt{x^3y}\sqrt[5]{xy^4}
\\\\=
(x^3y)^{1/2}(xy^4)^{1/5}
\\\\=
(x^3y)^{5/10}(xy^4)^{2/10}
\\\\=
\sqrt[10]{(x^3y)^5}\cdot\sqrt[10]{(xy^4)^2}
.\end{array}
Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[10]{(x^3y)^5}\cdot\sqrt[10]{(xy^4)^2}
\\\\=
\sqrt[10]{x^{15}y^5}\cdot\sqrt[10]{x^2y^8}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
\sqrt[10]{x^{15}y^5(x^2y^8)}
\\\\=
\sqrt[10]{x^{15+2}y^{5+8}}
\\\\=
\sqrt[10]{x^{17}y^{13}}
.\end{array}
Extracting the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[10]{x^{17}y^{13}}
\\\\=
\sqrt[10]{x^{10}y^{10}\cdot x^7y^3}
\\\\=
\sqrt[10]{(xy)^{10}\cdot x^7y^3}
\\\\=
xy\sqrt[10]{x^7y^3}
.\end{array}
Note that all variables are assumed to be positive.
