Question

Use the Quotient Property to simplify square roots.

(a) $\sqrt{\frac{75r^9}{s^8}}$

(b) $\sqrt[3]{\frac{54a^8}{b^3}}$

(c) $\sqrt[4]{\frac{64c^5}{d^4}}$

Short answer

(a)$\frac{5r^4\sqrt{3r}}{s^4}$

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

(c)$\frac{2\left|c\right|\sqrt[4]{4c}}{\left|d\right|}$

Step-by-step solution

Part (a) Step 1. Given information.

Given expression:

$\sqrt{\frac{75r^9}{s^8}}$.

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.

$$\begin{gathered} \sqrt{\frac{75r^9}{s^8}}=\sqrt{\frac{25r^8.3r}{s^8}} \\ =\frac{\sqrt{25r^8.3r}}{\sqrt{s^8}} \\ =\frac{\sqrt{{\left(5r^4\right)}^2}\sqrt{3r}}{\sqrt{{\left(s^4\right)}^2}} \\ =\frac{5\left|r^4\right|\sqrt{3r}}{\left|s^4\right|} \\ =\frac{5r^4\sqrt{3r}}{s^4} \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[3]{\frac{54a^8}{b^3}}$.

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.

$$\begin{gathered} \sqrt[3]{\frac{54a^8}{b^3}}=\sqrt[3]{\frac{27a^6.2a^2}{b^3}} \\ =\frac{\sqrt[3]{27a^6.2a^2}}{\sqrt[3]{b^3}} \\ =\frac{\sqrt[3]{{\left(3a^2\right)}^3}.\sqrt[3]{2a^2}}{\sqrt[3]{b^3}} \\ =\frac{3a^2\sqrt[3]{2a^2}}{\left|b\right|} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{64c^5}{d^4}}$

Part (c) Step 2. Write the concept.

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

$$\begin{gathered} \sqrt[4]{\frac{64c^3}{d^4}}=\sqrt[4]{\frac{16c^4.4c}{d^4}} \\ =\frac{\sqrt[4]{16c^4.4c}}{\sqrt[4]{d^4}} \\ =\frac{\sqrt[4]{{\left(16c\right)}^4}.\sqrt[4]{4c}}{\sqrt[4]{{\left(d\right)}^4}} \\ =\frac{2\left|c\right|\sqrt[4]{4c}}{\left|d\right|} \end{gathered}$$