Solve the following :$ \sqrt[4]{\sqrt[3]{2^1}}$

Given :

The given term is $ \sqrt[4]{\sqrt[3]{2^1}}$

To do :

We have to solve the given term.

Solution :

$ \sqrt[4]{\sqrt[3]{2^1}}$ 

$2^1 = 2$

So, $ \sqrt[4]{\sqrt[3]{2^1}} =  \sqrt[4]{\sqrt[3]{2}}$

We know that,

$\sqrt[n]{a} = a^ \frac{1}{n}$

$\sqrt[3]{2} =  2^ \frac{1}{3}$

$ \sqrt[4]{2^ \frac{1}{3}}$

$ \sqrt[4]{2^ \frac{1}{3}} = (2^ \frac{1}{3}) ^\frac{1}{4} $

We know that, $ (a^ \frac{1}{n}) ^\frac{1}{m} =  a^ \frac{1}{n \times m}$

$(2^ \frac{1}{3}) ^\frac{1}{4}  = 2^ \frac{1}{3 \times 4}$

 $=2^ \frac{1}{12} $

By reframing this, $\sqrt[n]{a} = a^ \frac{1}{n}$

We get , $a^ \frac{1}{n} =\sqrt[n]{a}$ 

Therefore,  $2^ \frac{1}{12} =\sqrt[12]{2}$ 

The value of $ \sqrt[4]{\sqrt[3]{2^1}}$ is $\sqrt[12]{2}$

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Updated on: 10-Oct-2022

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