Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$10x^2|y^{5}|\sqrt{2x}$
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}