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Q. 107

Expert-verifiedFound in: Page 772

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

Given expression:

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

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 $n\ge 2$.

Given radical expression:

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

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 $n\ge 2$.

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

Given expression:

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

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 $n\ge 2$.

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

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