Q. 8.45 - Intermediate Algebra

Question

Simplify:

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

Step-by-Step Solution

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, $\frac{2x^2 \sqrt[4]{5x^2}}{y}.$

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, 

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