Question

Use the Quotient Property to simplify square roots.

(a)$\sqrt{\frac{72x^5}{y^6}}$

(b)$\sqrt[5]{\frac{96r^{11}}{s^5}}$

(c)$\sqrt[6]{\frac{128u^7}{v^{12}}}$

Short answer

(a)$\frac{6x^2\sqrt{2x}}{\left|y^3\right|}$

(b)$\frac{2r^3\sqrt[5]{3r}}{\left|s\right|}$

(c)$\frac{2\left|u\right|\sqrt[6]{2u}}{\left|v^2\right|}$

Step-by-step solution

Part (a) Step 1. Given information.

Given expression:

$\sqrt{\frac{72x^5}{y^6}}$.

Part (a) Step 2. Write the concept.

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 $n\ge 2$.

Part (a) Step 3. Calculation

$$\begin{gathered} \sqrt{\frac{72x^5}{y^6}}=\sqrt{\frac{36x^4·2x}{y^6}} \\ =\frac{\sqrt{36x^4·2x}}{\sqrt{y^6}} \\ =\frac{6\left|x^2\right|\sqrt{2x}}{\left|y^3\right|} \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[5]{\frac{96r^{11}}{s^5}}$.

Part (b) Step 2. Write the concept.

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 $n\ge 2$.

Part (b) Step 3. Calculation

$$\begin{gathered} \sqrt[5]{\frac{96r^{11}}{s^5}}=\sqrt[5]{\frac{32r^{10}·3r}{s^5}} \\ =\frac{\sqrt[5]{32r^{10}·3r}}{\sqrt[5]{s^5}} \\ =\frac{2r^2\sqrt[5]{3r}}{\left|s\right|} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[6]{\frac{128u^7}{v^{12}}}$.

Part (c) Step 2. Write the concept.

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 $n\ge 2$.

Part (c) Step 3. Calculation

$$\begin{gathered} \sqrt[6]{\frac{128u^7}{v^{12}}}=\sqrt[6]{\frac{64u^6·2u}{v^{12}}} \\ =\frac{\sqrt[6]{64u^6·2u}}{\sqrt[6]{v^{12}}} \\ =\frac{\sqrt[6]{{\left(2u\right)}^6}·\sqrt[6]{2u}}{\sqrt[6]{{\left(v^2\right)}^6}} \\ =\frac{2\left|u\right|\sqrt[6]{2u}}{\left|v^2\right|} \end{gathered}$$