Question

Evaluate :
$\frac{ \sqrt{1296} }{2916}$
​

Answer 1

Answer:

$\frac{\sqrt{1296}}{2916}=\frac{1}{81}\quad \left(\mathrm{Decimal:\quad }\:0.01235\dots \right)$

Step-by-step explanation:

$\frac{\sqrt{1296}}{2916}$

$\black{\mathrm{Factor}\:\sqrt{1296}:}$

$\gray{\mathrm{Factor\:}1296=2^4\cdot \:3^4}$

$=\sqrt{2^4\cdot \:3^4}$

$\gray{\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}}$

$=\sqrt{2^4}\sqrt{3^4}$

$\gray{\mathrm{Simplify}\:\sqrt{2^4}:\quad 2^2}$

$\gray{\mathrm{Simplify}\:\sqrt{3^4}:\quad 3^2}$

$=2^2\cdot \:3^2$

$\black{\mathrm{Factor}\:2916:\quad 3^6\cdot \:2^2}$

$=\frac{2^2\cdot \:3^2}{3^6\cdot \:2^2}$

$\black{\mathrm{Cancel\:}\frac{2^2\cdot \:3^2}{2^2\cdot \:3^6}:}$

$\frac{2^2\cdot \:3^2}{2^2\cdot \:3^6}$

$\gray{\mathrm{Cancel\:the\:common\:factor:}\:2^2}$

$=\frac{3^2}{3^6}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=\frac{1}{x^{b-a}}}$

$\gray{\frac{3^2}{3^6}=\frac{1}{3^{6-2}}}$

$=\frac{1}{3^{6-2}}$

$\gray{\mathrm{Subtract\:the\:numbers:}\:6-2=4}$

$=\frac{1}{3^4}$

$\gray{3^4=81}$

$=\frac{1}{81}$