Simplify: (4 13 )^4 (13 7 )^2 (7 4 )^3 - Vidyadip

Question

Simplify: $\big(\frac{4}{13}\big)^4\times\big(\frac{13}{7}\big)^2\times\big(\frac{7}{4}\big)^3$

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Answer

$\big(\frac{4}{13}\big)^4\times\big(\frac{13}{7}\big)^2\times\big(\frac{7}{4}\big)^3$
$=\frac{(4)^4}{(13)^4}\times\frac{(13)^2}{(7)^2}\times\frac{(7)^3}{(4)^3}$
$=(4)^4\times(4)^{-3}\times(13)^2\times(13)^{-4}\times(7)^{3}\times(7)^{-2}$
$\big[\because\text{a}^{\text{-m}}=\frac{1}{\text{a}^{\text{m}}}\big]$
$=(4)^{4-3}\times(13)^{2-4}\times(7)^{3-2}$
$[\because\text{a}^{\text{m}}\times\text{a}^{\text{n}}=\text{a}^{\text{m}-\text{n}}]$
$=(4)^1\times(13)^{-2}\times(7)^1$
$=4\times\frac{1}{169}\times7$
$\big[\because\text{a}^{\text{-m}}=\frac{1}{\text{a}^{\text{m}}}\big]$
$=\frac{28}{169}$

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