Question

Simplify:

$\left(a\right)\sqrt{\frac{80m^3}{n^6}}\left(b\right)\sqrt[3]{\frac{108c^{10}}{d^6}}\left(c\right)\sqrt[4]{\frac{80x^{10}}{y^4}}.$

Short answer

Part a. After simplifying we get,$\frac{4m\sqrt{5m}}{n^3}.$

Part b. After simplifying we get,$\frac{3c^3\sqrt[3]{4c}}{d^2}.$

Part c. After simplifying we get,$\frac{2x^2\sqrt[4]{5x^2}}{y}.$

Step-by-step solution

Part a Step 1. Given Information 

The given expression is$\sqrt{\frac{80m^3}{n^6}}.$

We have to simplify it.

Part a Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals 

Thus, we write $\sqrt{\frac{80m^3}{n^6}}$as $\frac{\sqrt{80m^3}}{n^6}.$

Part a Step 3. Simplify the radicals of both the numerator and denominator 

After simplifying we get,

$$\begin{gathered} \frac{\sqrt{80m^3}}{\sqrt{n^6}} \\ =\frac{\sqrt{{\left(4m\right)}^2}·\sqrt{5m}}{\sqrt{{\left(n^3\right)}^2}} \\ =\frac{4m\sqrt{5m}}{n^3} \end{gathered}$$

Part b Step 1. Given Information

The given expression is$\sqrt[3]{\frac{108c^{10}}{d^6}}.$

We have to simplify it.

Part b Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals 

Thus, we write $\sqrt[3]{\frac{108c^{10}}{d^6}}$as $\frac{\sqrt[3]{108c^{10}}}{\sqrt[3]{d^6}}.$

Part b Step 3. Simplify the radicals of both the numerator and denominator

After simplifying we get,

$$\begin{gathered} \frac{\sqrt[3]{108c^{10}}}{\sqrt[3]{d^6}} \\ =\frac{\sqrt[3]{{\left(3c^3\right)}^3}·\sqrt[3]{4c}}{\sqrt[3]{{\left(d^2\right)}^3}} \\ =\frac{3c^3\sqrt[3]{4c}}{d^2} \end{gathered}$$

Part c Step 1. Given Information

The given expression is$\sqrt[4]{\frac{80x^{10}}{y^4}}.$

We have to simplify it.

Part c Step 2. Use the Quotient property to rewrite the radical as the quotient of two radicals 

Thus, we write$\sqrt[4]{\frac{80x^{10}}{y^4}}$as$\frac{\sqrt[4]{80x^{10}}}{\sqrt[4]{y^4}}.$

Part c Step 3. Simplify the radicals of both the numerator and denominator 

After simplifying we get,

$$\begin{gathered} \frac{\sqrt[4]{80x^{10}}}{\sqrt[4]{y^4}} \\ =\frac{\sqrt[4]{{\left(2x^2\right)}^4}·\sqrt[4]{5x^2}}{y} \\ =\frac{2x^2\sqrt[4]{5x^2}}{y} \end{gathered}$$