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Simplify the following :
$\frac{12 \sqrt[4]{15}}{8 \sqrt[3]{3}}$
Given :
The given expression is $\frac{12 \sqrt[4]{15}}{8 \sqrt[3]{3}}$.
To do :
We have to simplify the given expression.
Solution :
$\frac{12 \sqrt[4]{15}}{8 \sqrt[3]{3}}$
$\frac{12}{8} = \frac{3}{2}$
$\frac{3 \sqrt[4]{15}}{2 \sqrt[3]{3}}$
We know that,
$\sqrt[2]{a} = a^{\frac{1}{2}}$
Similarly,
$\sqrt[4]{15} = 15^{\frac{1}{4}}$
$\sqrt[3]{3} = 3^{\frac{1}{3}}$
$\frac{3 \sqrt[4]{15}}{2 \sqrt[3]{3}} = \frac{3}{2} \times \frac{ 15^{\frac{1}{4}}}{3^{\frac{1}{3}}}$
LCM of 3 , 4 is 12.
$ =\frac{3}{2} \times \frac{ 15^{\frac{1 \times 3}{4\times3}}}{3^{\frac{1\times4}{3\times4}}} $
$ =\frac{3}{2} \times \frac{ 15^{\frac{3}{12}}}{3^{\frac{4}{12}}} $
We know that, $a^\frac{m}{n} = (a^m)^\frac{1}{n}$
So, $ =\frac{3}{2} \times \frac{ (15^3)^{\frac{1}{12}}}{(3^4)^{\frac{1}{12}}} $
We know that, $\frac{a^m}{b^m} = (\frac{a}{b})^m$
$ =\frac{3}{2} \times (\frac{15^3}{3^4}) ^\frac{1}{12} $
$=\frac{3}{2} \times \sqrt[12]{\frac{15^3}{3^4}}$
$=\frac{3}{2} \times \sqrt[12]{\frac{15\times15\times15}{3\times3\times3\times3}}$
$=\frac{3}{2} \times \sqrt[12]{\frac{5\times5\times5}{3}}$
$=\frac{3}{2} \times \sqrt[12]{\frac{125}{3}}$
$\frac{12 \sqrt[4]{15}}{8 \sqrt[3]{3}} =\frac{3}{2} \sqrt[12]{\frac{125}{3}}$