Question

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$$\begin{gathered} a.\sqrt{\frac{72}{98}} \\ b.\sqrt[3]{\frac{24}{81}} \\ c.\sqrt[4]{\frac{6}{96}} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.\frac{6}{7}. \\ b.\frac{2}{3}. \\ c.\frac{1}{2}. \end{gathered}$$

Step-by-step solution

Given expressions are,

$$\begin{gathered} a.\sqrt{\frac{72}{98}} \\ b.\sqrt[3]{\frac{24}{81}} \\ c.\sqrt[4]{\frac{6}{96}} \end{gathered}$$

The objective is, we need to find the simplified expressions for the given expressions.

The quotient property rule is,

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$.

Part a Step 2:7298.

Now use the quotient property rule.

$$\begin{gathered} \sqrt{\frac{72}{98}}=\frac{\sqrt{72}}{\sqrt{98}} \\ =\frac{\sqrt{2\times 36}}{\sqrt{2\times 49}} \\ =\frac{\sqrt{36}}{\sqrt{49}} \\ =\frac{\sqrt{{\left(6\right)}^2}}{\sqrt{{\left(7\right)}^2}} \\ =\frac{6}{7} \end{gathered}$$

Part b Step 2:24813 .

Now use the quotient property rule.

$$\begin{gathered} \sqrt[3]{\frac{24}{81}}=\frac{\sqrt[3]{24}}{\sqrt[3]{81}} \\ =\frac{\sqrt[3]{3\times 8}}{\sqrt[3]{3\times 27}} \\ =\frac{\sqrt[3]{8}}{\sqrt[3]{27}} \\ =\frac{2}{3} \end{gathered}$$

Part c Step 2: 6964. 

Now use the quotient rule.

$$\begin{gathered} \sqrt[4]{\frac{6}{96}}=\frac{\sqrt[4]{6}}{\sqrt[4]{96}} \\ =\frac{\sqrt[4]{2\times 3}}{\sqrt[4]{16\times 2\times 3}} \\ =\frac{\sqrt[4]{1}}{\sqrt[4]{16}} \\ =\frac{1}{2} \end{gathered}$$