Evaluate: (81 16 )^ -3 4 [ (25 9 )^ -3 2 (5 2 )^ -3 ] - Vidyadip

Question

Evaluate:
$\Big(\frac{81}{16}\Big)^{-\frac{3}{4}}\Big[\Big(\frac{25}{9}\Big)^{-\frac{3}{2}}\div\Big(\frac{5}{2}\Big)^{-3}\Big]$

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Answer

$\Big(\frac{81}{16}\Big)^{-\frac{3}{4}}\Big[\Big(\frac{25}{9}\Big)^{-\frac{3}{2}}\div\Big(\frac{5}{2}\Big)^{-3}\Big]$
$=\Big(\frac{16}{81}\Big)^{\frac{3}{4}}\Big[\Big(\frac{9}{25}\Big)^{\frac{3}{2}}\div\Big(\frac{2}{5}\Big)^{3}\Big]$
$=\Big(\frac{2^4}{3^4}\Big)^{\frac{3}{4}}\Big[\Big(\frac{3^2}{5^2}\Big)^{\frac{3}{2}}\div\frac{2^3}{5^3}\Big]$
$=\frac{2^{4\times\frac{3}{4}}}{3^{4\times\frac{3}{4}}}\Bigg[\frac{3^{2\times\frac{3}{2}}}{5^{2\times\frac{3}{2}}}\div\frac{8}{125}\Bigg]$
$=\frac{2^3}{3^3}\Big[\frac{3^3}{5^3}\times\frac{125}{8}\Big]$
$=\frac{8}{27}\Big[\frac{27}{125}\times\frac{125}{8}\Big]$
$=\frac{8}{27}\times\frac{27}{8}$
$=1$

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