Find the value of
Question:

Find the value of $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$.

Solution:

We have $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$

$=\frac{4}{\left(6^{3}\right)^{-\frac{2}{3}}}+\frac{1}{\left(16^{2}\right)^{-\frac{3}{4}}}+\frac{2}{\left(3^{5}\right)^{-\frac{1}{5}}}$

$=\frac{4}{6^{3 \times\left(-\frac{2}{3}\right)}}+\frac{1}{16^{2 \times\left(-\frac{3}{4}\right)}}+\frac{2}{3^{5 \times\left(-\frac{1}{5}\right)}}$     $\left[\because\left(a^{m n}\right)^{n}=a^{m n}\right]$

$=\frac{4}{6^{-2}}+\frac{1}{16^{-\frac{3}{2}}}+\frac{2}{3^{-1}}$

$=4 \times 6^{2}+16^{\frac{3}{2}}+2 \times 3^{1}$          $\left[\because \frac{1}{a}=a^{-1}\right]$

$=4 \times 36+\left((4)^{2}\right)^{3 / 2}+2 \times 3^{1}$

$=4 \times 36+4^{3}+6$

$=144+64+6=214$

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