Use the Quotient Property to simplify square roots.

(a) $\sqrt{\frac{50r^5{s}^2}{128r^2{s}^6}}$

(b) $\sqrt[3]{\frac{24m^9{n}^7}{375m^{4}n}}$

(c) $\sqrt[4]{\frac{81m^2{n}^8}{256m{n}^2}}$

Given expression:

$\sqrt{\frac{50r^5{s}^2}{128r^2{s}^6}}$.

For simplifying the radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a}and\sqrt[n]{b}$are the real number, $b\ne 0$for any integer$n\ge 2.$

$$\begin{gathered} \sqrt{\frac{50r^5{s}^2}{128r^2{s}^6}}=\sqrt{\frac{50}{128}.r^{5-2}.\frac{1}{s^{6-2}}} \\ =\sqrt{\frac{25}{64}.r^3.\frac{1}{s^4}} \\ =\sqrt{{\left(\frac{5}{8}\right)}^2.r^2.\frac{1}{{\left(s^2\right)}^2}.r} \\ =\frac{5}{8}\left|r\right|\frac{1}{\left|s^2\right|}\sqrt{r} \\ =\frac{5\left|r\right|\sqrt{r}}{8s^2} \end{gathered}$$

Given radical expression:

$\sqrt[3]{\frac{24m^9{n}^7}{375m^{4}n}}$.

For simplifying the radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a}and\sqrt[n]{b}$are the real number, $b\ne 0$for any integer $n\ge 2$.

$$\begin{gathered} \sqrt[3]{\frac{24m^9{n}^7}{375m^{4}n}}=\sqrt[3]{\frac{24}{375}.m^{9-4}.n^{7-4}} \\ =\sqrt[3]{\frac{8}{125}.m^5.n^3} \\ =\sqrt[3]{{\left(\frac{2}{5}\right)}^3.m^3.n^3.m^2} \\ =\frac{2}{5}mn\sqrt[3]{m^2} \end{gathered}$$

Given expression:

$\sqrt[4]{\frac{81m^2{n}^8}{256m{n}^2}}$.

radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a}and\sqrt[n]{b}$are the real number, $b\ne 0$for any integer $n\ge 2$.

$$\begin{gathered} \sqrt[4]{\frac{81m^2{n}^8}{256m{n}^2}}=\sqrt[4]{\frac{81}{256}.m^{2-1}.n^{8-2}} \\ =\sqrt[4]{\frac{3}{4}.m.n^6} \\ =\sqrt[]{{\left(\frac{3}{4}\right)}^4}.\sqrt[4]{m}.\sqrt[4]{n^4}.\sqrt[4]{n^2} \\ =\frac{3}{4}\left|n\right|\sqrt[4]{mn} \end{gathered}$$