Question

Use the Quotient Property to simplify square roots.

(a)$\frac{\sqrt{80q^5}}{\sqrt{5q}}$

(b)$\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}$

(c) $\frac{\sqrt[4]{80m^7}}{\sqrt[4]{5m}}$

Short answer

(a)$4q^2$

(b)$-5$

(c) $\left|2m\right|\sqrt[4]{m^2}$

Step-by-step solution

Part (a) Step 1. Given information.

Given expression:

$\frac{\sqrt{80q^5}}{\sqrt{5q}}$.

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

$$\begin{gathered} \frac{\sqrt{80q^5}}{\sqrt{5q}}=\sqrt{\frac{80q^5}{5q}} \\ =\sqrt{\frac{16.5.q^4.q}{5.q}} \\ =\sqrt{16.q^4} \\ =4\sqrt{{\left(q^2\right)}^2} \\ =4q^2 \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}$.

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

$$\begin{gathered} \frac{\sqrt[3]{-625}}{\sqrt[3]{5}}=\sqrt[3]{\frac{-625}{5}} \\ =\sqrt[3]{\frac{-125.5}{5}} \\ =\sqrt[3]{{(-5)}^3} \\ =-5 \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\frac{\sqrt[4]{80m^7}}{\sqrt[4]{5m}}$.

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

$$\begin{gathered} \frac{\sqrt[4]{80m^7}}{\sqrt[4]{5m}}=\sqrt[4]{\frac{80m^7}{5m}} \\ =\sqrt[4]{\frac{16.5.m^4.m^3}{5.m}} \\ =\sqrt[4]{16.m^4.m^2} \\ =\left|2m\right|\sqrt[4]{m^2} \end{gathered}$$