Simplify:
$ \left(\frac{81}{16}\right)^{\frac{-3}{4}} \times\left[\left(\frac{25}{4}\right)^{\frac{-3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right] $

Given:

\( \left(\frac{81}{16}\right)^{\frac{-3}{4}} \times\left[\left(\frac{25}{4}\right)^{\frac{-3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right] \)

To do:

We have to simplify the given expression.
Solution:

We know that,

$(a^m)^n=(a)^{m\times n}$

$a^m \times a^n=a^{(m+n)}$
$a^m \div a^n=a^{(m-n)}$
 $a^m \times b^m=(a\times b)^m$

$\frac{a^m}{b^m}=(\frac{a}{b})^m$

Therefore,

$\left(\frac{81}{16}\right)^{\frac{-3}{4}} \times\left[\left(\frac{25}{4}\right)^{\frac{-3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right]=(\frac{3^4}{2^4})^{\frac{-3}{4}} \times[(\frac{5^2}{2^2})^{\frac{-3}{2}} \times (\frac{5}{2})^3]$

$=(\frac{3}{2})^{4\times\frac{-3}{4}} \times[(\frac{5}{2})^{2\times\frac{-3}{2}} \times (\frac{5}{2})^3]$

$=(\frac{3}{2})^{-3} \times[(\frac{5}{2})^{2\times\frac{-3}{2}} \times (\frac{5}{2})^3]$

$=(\frac{2}{3})^3\times[(\frac{2}{5})^3\times(\frac{5}{2})^3]$

$=(\frac{2}{3})^3\times[(\frac{2\times5}{5\times2})^3]$

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

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

$=\frac{8}{27}$

Hence, \( \left(\frac{81}{16}\right)^{\frac{-3}{4}} \times\left[\left(\frac{25}{4}\right)^{\frac{-3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right] \)$=\frac{8}{27}$.

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Updated on: 10-Oct-2022

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