Question

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

$$\begin{gathered} a.\sqrt{\frac{121}{16}} \\ b.\sqrt[3]{\frac{16}{250}} \\ c.\sqrt[4]{\frac{32}{162}} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.\frac{11}{4} \\ b.\frac{2}{5} \\ c.\frac{2}{3} \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} a.\sqrt{\frac{121}{16}} \\ b.\sqrt[3]{\frac{16}{250}} \\ c.\sqrt[4]{\frac{32}{162}} \end{gathered}$$

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

The quotient property is given by,

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

Part a Step 2 src="data:image/svg+xml;base64,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" localid="1645431102703" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/91920281-5371-48c0-9ac9-ce8d7d48d2d8.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155617Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=3a1d1d96aaf7b0fe15613e4813781ff0195c0f682c85619a994dda3044381caa" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/91920281-5371-48c0-9ac9-ce8d7d48d2d8.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T081319Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=ac08d6dea2a1a1ce2bf82e55a5e76279913877860b97b6c96925e9402dd6e064" 12116.

Now use the quotient property.

$$\begin{gathered} \sqrt{\frac{121}{16}}=\sqrt{\frac{11\times 11}{4\times 4}} \\ =\sqrt{\frac{{\left(11\right)}^2}{{\left(4\right)}^2}} \\ =\frac{\sqrt{{\left(11\right)}^2}}{\sqrt{{\left(4\right)}^2}} \\ =\frac{11}{4} \end{gathered}$$

Part b Step 2 162503.

$$\begin{gathered} \sqrt[3]{\frac{16}{250}}=\sqrt[3]{\frac{2\times 8}{2\times 125}} \\ =\sqrt[3]{\frac{8}{125}} \end{gathered}$$

Now use the quotient property.

$$\begin{gathered} \sqrt[3]{\frac{16}{250}}=\frac{\sqrt[3]{8}}{\sqrt[3]{125}} \\ =\frac{\sqrt[3]{{\left(2\right)}^3}}{\sqrt[3]{{\left(5\right)}^3}} \\ =\frac{2}{5} \end{gathered}$$

Part c Step 2321624.

$$\begin{gathered} \sqrt[4]{\frac{32}{162}}=\sqrt[4]{\frac{2\times 16}{2\times 81}} \\ =\sqrt[4]{\frac{16}{81}} \end{gathered}$$

Now use the quotient property.

$$\begin{gathered} \sqrt[4]{\frac{32}{162}}=\frac{\sqrt[4]{16}}{\sqrt[4]{81}} \\ =\frac{\sqrt[4]{{\left(2\right)}^4}}{\sqrt[4]{\left(3^4\right)}} \\ =\frac{2}{3} \end{gathered}$$