The value of $ \frac{\left((243)^{1 / 5}\right)^{4}}{\left((32)^{1 / 5}\right)^{4}}=? $
A. $ \quad \frac{3}{2} $
B. $ \left(\frac{3}{2}\right)^{-4} $
C. $ \frac{1}{2^{-4} \times 3^{-4}} $
D. $ \frac{1}{2^{4} \times 3^{-4}} $

Given:

\( \frac{\left((243)^{1 / 5}\right)^{4}}{\left((32)^{1 / 5}\right)^{4}} \)

To do:

We have to find the value of \( \frac{\left((243)^{1 / 5}\right)^{4}}{\left((32)^{1 / 5}\right)^{4}} \).

Solution:

We know that,

$(a^m)^n=a^{m\times n}$ $\frac{a^m}{b^m}=(\frac{a}{b})^m$  Therefore,

$\frac{\left((243)^{1 / 5}\right)^{4}}{\left((32)^{1 / 5}\right)^{4}}= \frac{\left((3^5)^{1 / 5}\right)^{4}}{\left((2^5)^{1 / 5}\right)^{4}}$

$=\frac{((3)^{5\times\frac{1}{5}})^4}{((2)^{5\times\frac{1}{5}})^4}$

$=\frac{(3)^{1\times4}}{(2)^{1\times4}}$

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

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

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

$=\frac{1}{3^{-4}\times2^4}$.

Option D is the correct answer.