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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 8: Q. 87 (page 772)

Use the Quotient Property to Simplify Radical Expressions.
In the following exercises, use the Quotient Property to simplify square roots.
$a . \sqrt{\frac{45}{80}} b . \sqrt[3]{\frac{8}{27}} c . \sqrt[4]{\frac{1}{81}}$

Short Answer

Expert verified

The simplified expressions are,

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

Step by step solution

01

Step 1 Given expressions are,

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

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

The quotient property is given as,

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

02

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" 4580

$\sqrt{\frac{45}{80}} = \sqrt{\frac{9 脳 5}{16 脳 5}} = \sqrt{\frac{9}{16}} = \sqrt{\frac{{(3)}^{2}}{{(4)}^{2}}}$

Now use the quotient property rule.

$\sqrt{\frac{45}{80}} = \frac{\sqrt{{(3)}^{2}}}{\sqrt{{(4)}^{2}}} = \frac{3}{4}$

03

Part b Step 2 8273 .

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

04

Part c Step 2 1814

Now use the quotient property rule.

$\sqrt[4]{\frac{1}{81}} = \frac{\sqrt[4]{1}}{\sqrt[4]{81}} = \frac{\sqrt[4]{{(1)}^{4}}}{\sqrt[4]{{(3)}^{4}}} = \frac{1}{3}$

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Most popular questions from this chapter

In the following exercises, simplify.

(a) $64^{\frac{5}{2}}$

(b) $81^{\frac{- 3}{2}}$

(c) ${(- 27)}^{\frac{2}{3}}$

In the following exercises, simplify using absolute value signs as needed. $\sqrt{147 m^{7} n^{11}}, \sqrt[3]{48 x^{6} y^{7}}, \sqrt[4]{32 x^{5} y^{4}}$

In the following exercises, solve.

$\sqrt{5 v - 2} + 5 = 0$

In the following exercises, simplify.

(a) $- 64^{\frac{3}{2}}$

(b) $- 64^{- \frac{3}{2}}$

(c) ${(- 64)}^{\frac{3}{2}}$

Simplify:

a. $x^{\frac{1}{6}} 路 x^{\frac{4}{3}}$ b. ${(x^{6})}^{\frac{4}{3}}$ c. $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$

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