# Chapter 13 - Conic Sections - 13.1 Conic Sections: Parabolas and Circles - 13.1 Exercise Set - Page 855: 69

$10{{x}^{2}}\sqrt{w}$

#### 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 $10{{x}^{2}}\sqrt{w}$

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