Question

Use the Quotient Property to simplify square roots.

(a)$\frac{\sqrt{50m^7}}{\sqrt{2m}}$

(b)$\sqrt[3]{\frac{1250}{2}}$

(c)$\sqrt[4]{\frac{486y^9}{2y^3}}$

Short answer

(a)$5\left|m^3\right|$

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

(c) $3\left|y\right|\sqrt[4]{3y^2}$

Step-by-step solution

Part (a) Step 1. Given information.

Given expression:

$\frac{\sqrt{50m^7}}{\sqrt{2m}}$.

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 $\sqrt[n]{a},\sqrt[n]{b}$are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (a) Step 3. Calculation

$$\begin{gathered} \frac{\sqrt{50m^7}}{\sqrt{2m}}=\sqrt{\frac{50m^7}{2m}} \\ =\sqrt{\frac{25·2·m^6·m}{2·m}} \\ =\sqrt{25·m^6} \\ =5\sqrt{{\left(m^3\right)}^2} \\ =5\left|m^3\right| \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[3]{\frac{1250}{2}}$.

Part (b) Step 2. Write the concept.

For simplifying the radical expression following formula is used:

if $$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a},\sqrt[n]{b}$are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 3. Calculation.

$$\begin{gathered} \sqrt[3]{\frac{1250}{2}}=\sqrt[3]{\frac{125·5·2}{2}} \\ =\sqrt[3]{125·5} \\ =\sqrt[3]{{\left(5\right)}^3·5} \\ =5\sqrt[3]{5} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{486y^9}{2y^3}}$.

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 $\sqrt[n]{a},\sqrt[n]{b}$are the real number,$b\ne 0$for any integer $n\ge 2$.

Part (c) Step 3. Calculation

$$\begin{gathered} \sqrt[4]{\frac{486y^9}{2y^3}}=\sqrt[4]{\frac{81·3·2y^6·y^3}{2·y^3}} \\ =\sqrt[4]{\frac{81·3·2·y^6}{2}} \\ =\sqrt[4]{81·3·y^4·y^2} \\ =3\left|y\right|\sqrt[4]{3y^2} \end{gathered}$$