91Ó°ÊÓ

Given expression:

$\frac{\sqrt{72n^{11}}}{\sqrt{2n}}$.

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,$b≠0$for any integer$nâ‰\yen 2$.

$$\begin{gathered} \frac{\sqrt{72n^{11}}}{\sqrt{2n}}=\sqrt{\frac{72n^{11}}{2n}} \\ =\sqrt{\frac{36.2.n^{10}.n}{2.n}} \\ =\sqrt{36.n^{10}} \\ =\sqrt{36}.\sqrt{n^{10}} \\ =\sqrt{6^2.{\left(n^5\right)}^2} \\ =6\left|n^5\right| \end{gathered}$$

Given radical expression:

$\sqrt[3]{\frac{162}{6}}$.

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, $b≠0$for any integer $nâ‰\yen 2$.

$$\begin{gathered} \sqrt[3]{\frac{162}{3}}=\sqrt[3]{\frac{27.6}{6}} \\ =\sqrt[3]{27} \\ =3 \end{gathered}$$

Given expression:

$\sqrt[4]{\frac{160r^{10}}{5r^3}}$.

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, $b≠0$for any integer $nâ‰\yen 2$.

$$\begin{gathered} \sqrt[4]{\frac{160r^{10}}{5r^3}}=\sqrt[4]{\frac{16.2.5.r^{10}}{5.r^3}} \\ =\sqrt[4]{16.2.r^2} \\ =\sqrt[4]{2^4.2r^4.y^3} \\ =3\left|r\right|\sqrt[4]{2r^3} \end{gathered}$$