Answer
$10x^2|y^{5}|\sqrt{2x}$
Work Step by Step
Extracting the factor that is a perfet power of the index, then
\begin{array}{l}\require{cancel}
\sqrt{200x^5y^{10}}
\\\\=
\sqrt{100x^4y^{10}\cdot2x}
\\\\=
\sqrt{(10x^2y^{5})^2\cdot2x}
.\end{array}
Using $\sqrt[n]{x^n}=|x|$ if $n$ is even and $\sqrt[n]{x^n}=x$ if $n$ is odd, then
\begin{array}{l}\require{cancel}
\sqrt{(10x^2y^{5})^2\cdot2x}
\\\\=
|10x^2y^{5}|\sqrt{2x}
\\\\=
|10|\cdot|x^2|\cdot|y^{5}|\sqrt{2x}
\\\\=
10x^2|y^{5}|\sqrt{2x}
.\end{array}
