Q. 89 - Intermediate Algebra

Question

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

$$\begin{gathered} a. \sqrt{\frac{100}{36}} \\ b. \sqrt[3]{\frac{81}{375}} \\ c. \sqrt[4]{\frac{1}{256}} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expressions are,

$$\begin{gathered} a. \frac{5}{3} \\ b. \frac{3}{5} \\ c. \frac{1}{4} \end{gathered}$$

Step 1 Given expressions are,

$$\begin{gathered} a. \sqrt{\frac{100}{36}} \\ b. \sqrt[3]{\frac{81}{375}} \\ c. \sqrt[4]{\frac{1}{256}} \end{gathered}$$

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="1645430757272" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/5eee5792-e664-42e4-a96f-0b47edb2710b.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155613Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=118c2f38f09069172e70d3f0726abaf0dde506a9fa4770358460ff8801e52f49" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/5eee5792-e664-42e4-a96f-0b47edb2710b.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T080732Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=7a30c99e887d2a6a3c0c06faf07345921d0199d11b5a716398bd3be0c0fa4e9d" 100 36

Now use the quotient property.

$$\begin{gathered} \sqrt{\frac{100}{36}}=\frac{\sqrt{100}}{\sqrt{36}} \\ = \frac{\sqrt{10\times 10}}{\sqrt{6\times 6}} \\ =\frac{10}{6} \\ =\frac{5}{3} \end{gathered}$$

Part b Step 2 81 375 3

Now use the quotient property.$$\begin{gathered} \sqrt[3]{\frac{81}{375}}=\frac{\sqrt[3]{81}}{\sqrt[3]{375}} \\ =\frac{\sqrt[3]{3\times 27}}{\sqrt[3]{3\times 125}} \\ =\frac{\sqrt[3]{3\times 27}}{\sqrt[3]{3\times 125}} \\ =\frac{\sqrt[3]{27}}{\sqrt[3]{125}} \\ =\frac{3}{5} \end{gathered}$$

Part c Step 2   1 256 4   .

Now use the quotient property.

$$\begin{gathered} \sqrt[4]{\frac{1}{256}}=\frac{\sqrt[4]{1}}{\sqrt[4]{256}} \\ = \frac{\sqrt[4]{{\left(1\right)}^4}}{\sqrt[4]{{\left(4\right)}^4}} \\ =\frac{1}{4} \end{gathered}$$