Q. 87 - Intermediate Algebra

Question

Use the Quotient Property to Simplify Radical Expressions.

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

$$\begin{gathered} a. \sqrt{\frac{45}{80}} \\ b. \sqrt[3]{\frac{8}{27}} \\ c. \sqrt[4]{\frac{1}{81}} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expressions are,

$$\begin{gathered} a. \frac{3}{4} \\ b. \frac{2}{3} \\ c. \frac{1}{3} \end{gathered}$$

Step 1 Given expressions are,

$$\begin{gathered} a. \sqrt{\frac{45}{80}} \\ b. \sqrt[3]{\frac{8}{27}} \\ c. \sqrt[4]{\frac{1}{81}} \end{gathered}$$

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

The quotient property is given as,

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

Part a Step 2 src="data:image/svg+xml;base64,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" localid="1645429255625" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/b7ab296a-3186-4b22-a7ad-7bab2487e45d.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155604Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=b89999c988840397d2e81b458c4f49b0eabdf86f5d8ddaa49b1c788bbe68fca3" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/b7ab296a-3186-4b22-a7ad-7bab2487e45d.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T074241Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=30cf7ff6a496759ea7ae393584827e9eb3efaa0eef169dbb4672035c7c6368a1"   45 80  

$$\begin{gathered} \sqrt{\frac{45}{80}} =\sqrt{\frac{9\times 5}{16\times 5}} \\ =\sqrt{\frac{9}{16}} \\ =\sqrt{\frac{{\left(3\right)}^2}{{\left(4\right)}^2}} \end{gathered}$$

Now  use the quotient property rule.

$$\begin{gathered} \sqrt{\frac{45}{80}}=\frac{\sqrt{{\left(3\right)}^2}}{\sqrt{{\left(4\right)}^2}} \\ =\frac{3}{4} \end{gathered}$$

Part b Step 2 8 27 3   .

Now use the quotient property rule.$$\begin{gathered} \sqrt[3]{\frac{8}{27}} =\frac{\sqrt[3]{8}}{\sqrt[3]{27}} \\ =\frac{\sqrt[3]{{\left(2\right)}^3}}{\sqrt[3]{{\left(3\right)}^3}} \\ =\frac{2}{3} \end{gathered}$$

Part c Step 2   1 81 4

Now use the quotient property rule.

$$\begin{gathered} \sqrt[4]{\frac{1}{81}}=\frac{\sqrt[4]{1}}{\sqrt[4]{81}} \\ =\frac{\sqrt[4]{{\left(1\right)}^4}}{\sqrt[4]{{\left(3\right)}^4}} \\ =\frac{1}{3} \end{gathered}$$