Answer
$10{{x}^{2}}\sqrt{w}$
Work Step by Step
Consider expression $\frac{\sqrt{200{{x}^{4}}{{w}^{2}}}}{\sqrt{2w}}$,
Apply quotient rule for radicals,
$\frac{\sqrt{200{{x}^{4}}{{w}^{2}}}}{\sqrt{2w}}=\sqrt{\frac{200{{x}^{4}}{{w}^{2}}}{2w}}$
Simplify the expression $\sqrt{\frac{200{{x}^{4}}{{w}^{2}}}{2w}}$,
$\sqrt{100{{x}^{4}}w}$
Identify the largest perfect square power factor,
$\sqrt{100{{x}^{4}}w}=\sqrt{{{\left( 10 \right)}^{2}}\times {{\left( {{x}^{2}} \right)}^{2}}}\times \sqrt{w}$
Find the square root. We assume $x\ge 0$,
$10{{x}^{2}}\sqrt{w}$
Partial check:
$\begin{matrix}
{{\left( 10{{x}^{2}}\sqrt{w} \right)}^{2}}\overset{?}{\mathop{=}}\,{{\left( 10{{x}^{2}} \right)}^{2}}{{\left( \sqrt{w} \right)}^{2}} \\
=100{{x}^{4}}w \\
\end{matrix}$
Thus, the expression $\frac{\sqrt{200{{x}^{4}}{{w}^{2}}}}{\sqrt{2w}}$ can be simplified as $10{{x}^{2}}\sqrt{w}$
