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} $

$=\frac{3}{2} \times \sqrt[12]{\frac{15^3}{3^4}}$

$=\frac{3}{2} \times \sqrt[12]{\frac{15\times15\times15}{3\times3\times3\times3}}$

$=\frac{3}{2} \times \sqrt[12]{\frac{5\times5\times5}{3}}$

$=\frac{3}{2} \times \sqrt[12]{\frac{125}{3}}$

$\frac{12 \sqrt[4]{15}}{8 \sqrt[3]{3}} =\frac{3}{2}  \sqrt[12]{\frac{125}{3}}$

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Updated on: 10-Oct-2022

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