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Q. 107

Expert-verified
Found in: Page 772

Intermediate Algebra

Book edition OER 2017
Author(s) OPENSTAX
Pages 1346 pages
ISBN 9780998625720

Use the Quotient Property to simplify square roots.(a) $\sqrt{\frac{32{x}^{5}{y}^{3}}{18{x}^{3}y}}$(b) $\sqrt[3]{\frac{5{x}^{6}{y}^{9}}{40{x}^{5}{y}^{3}}}$(c) $\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}$

(a) $\frac{4\left|xy\right|}{3}$

(b) $\frac{{y}^{2}\sqrt[3]{x}}{2}$

(c) $\frac{\left|ab\right|\sqrt[4]{a}}{2}$

See the step by step solution

Part (a) Step 1. Given information.

Given expression:

$\sqrt{\frac{32{x}^{5}{y}^{3}}{18{x}^{3}y}}$.

Part (a) Step 2. Write the concept.

For simplifying the radical expression following formula is used:

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and\phantom{\rule{0ex}{0ex}}\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

if $\sqrt[n]{a}and\sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 1. Given information.

$\sqrt[3]{\frac{5{x}^{6}{y}^{9}}{40{x}^{5}{y}^{3}}}.$

Part (b) Step 2. Write the concept.

For simplifying the radical expression following formula is used:

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and\phantom{\rule{0ex}{0ex}}\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

if $\sqrt[n]{a}and\sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 3. Calculation

$\sqrt[3]{\frac{5{x}^{6}{y}^{9}}{40{x}^{5}{y}^{3}}}=\sqrt[3]{\frac{5{x}^{5}.x.{y}^{3}.{y}^{6}}{40{x}^{5}{y}^{3}}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{1}{8}.x.{y}^{6}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{{\left({y}^{2}\right)}^{3}}{{2}^{2}}}.\sqrt[3]{x}\phantom{\rule{0ex}{0ex}}=\frac{{y}^{2}}{2}\sqrt[3]{x}$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}$.

Part (c) Step 2. Write the concept.

radical expression following formula is used:

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and\phantom{\rule{0ex}{0ex}}\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

if $\sqrt[n]{a}and\sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (c) Step 3. Calculation

$\sqrt[4]{\frac{5{a}^{8}{b}^{6}}{80{a}^{3}{b}^{2}}}=\sqrt[4]{\frac{5{a}^{5}.{a}^{3}.{b}^{4}.{b}^{2}}{80{a}^{3}{b}^{2}}}\phantom{\rule{0ex}{0ex}}=\sqrt[4]{\frac{1.{a}^{5}.{b}^{4}}{16}}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\left|ab\right|\sqrt[4]{a}$