Question

Use the Quotient Property to simplify square roots.

(a)$\sqrt{\frac{32x^5{y}^3}{18x^{3}y}}$

(b)$\sqrt[3]{\frac{5x^6{y}^9}{40x^5{y}^3}}$

(c) $\sqrt[4]{\frac{5a^8{b}^6}{80a^3{b}^2}}$

Short answer

(a) $\frac{4\left|xy\right|}{3}$

(b)$\frac{y^2\sqrt[3]{x}}{2}$

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

Step-by-step solution

Part (a) Step 1. Given information.

Given expression:

$\sqrt{\frac{32x^5{y}^3}{18x^{3}y}}$.

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. Calculation32x5y318x3y=32.x2.x3.y2.y18x3y=16x2y29=16242xy2=43xy

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[3]{\frac{5x^6{y}^9}{40x^5{y}^3}}.$

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

$$\begin{gathered} \sqrt[3]{\frac{5x^6{y}^9}{40x^5{y}^3}}=\sqrt[3]{\frac{5x^5.x.y^3.y^6}{40x^5{y}^3}} \\ =\sqrt[3]{\frac{1}{8}.x.y^6} \\ =\sqrt[3]{\frac{{\left(y^2\right)}^3}{2^2}}.\sqrt[3]{x} \\ =\frac{y^2}{2}\sqrt[3]{x} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{5a^8{b}^6}{80a^3{b}^2}}$.

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

$$\begin{gathered} \sqrt[4]{\frac{5a^8{b}^6}{80a^3{b}^2}}=\sqrt[4]{\frac{5a^5.a^3.b^4.b^2}{80a^3{b}^2}} \\ =\sqrt[4]{\frac{1.a^5.b^4}{16}} \\ =\frac{1}{2}\left|ab\right|\sqrt[4]{a} \end{gathered}$$