Question
Evaluate: $\frac{3^{4}\times12^{3}\times36}{2^{5}\times6^{3}}$
✓
Answer
We have, $\frac{3^{4}\times12^{3}\times36}{2^{5}\times6^{3}}$$=\frac{3^{4}\times(2^{2}\times3)^{3}\times(2^{2}\times3^{2})}{2^{5}\times(2\times3)^{3}}$ $\Big[\because 12=2\times2\times3\text{ and }36=2\times2\times3\times3\Big]$
$=\frac{3^{4}\times2^{6}\times3^{3}\times2^{2}\times3^{2}}{2^{5}\times2^{3}\times3^{3}}$ $\Big[\because(\text{a}\times\text{b})^{ \text{m}}=\text{a}^{\text{m}}\times\text{b}^{\text{m}}\Big]$
$=\frac{(3^{4}\times3^{2}\times3^{3})\times(2 ^{6}\times2^{2})}{(2^{5}\times2^{3})\times3^{3}}$
$\frac{3^{4+2+3}\times2^{6+2}}{2^{5+3}\times3^{3}}$ $\Big[\because\text{a}^{\text{m}}\times\text{a}^{\text{n}}=\text{a}^{\text{m+n}}\Big]$
$\frac{3^{9}\times2^{8}}{3^{3}\times2^{8}}=3^{9-3}\times2^{8-8}$ $\Big[\because\frac{\text{a}^{\text{m}}}{\text{a}^{\text{n}}}=\text{a}^{\text{m-n}}\Big]$
$=3^{6}\times2^{0}=3^{6}\times1=729$
$=\frac{3^{4}\times2^{6}\times3^{3}\times2^{2}\times3^{2}}{2^{5}\times2^{3}\times3^{3}}$ $\Big[\because(\text{a}\times\text{b})^{ \text{m}}=\text{a}^{\text{m}}\times\text{b}^{\text{m}}\Big]$
$=\frac{(3^{4}\times3^{2}\times3^{3})\times(2 ^{6}\times2^{2})}{(2^{5}\times2^{3})\times3^{3}}$
$\frac{3^{4+2+3}\times2^{6+2}}{2^{5+3}\times3^{3}}$ $\Big[\because\text{a}^{\text{m}}\times\text{a}^{\text{n}}=\text{a}^{\text{m+n}}\Big]$
$\frac{3^{9}\times2^{8}}{3^{3}\times2^{8}}=3^{9-3}\times2^{8-8}$ $\Big[\because\frac{\text{a}^{\text{m}}}{\text{a}^{\text{n}}}=\text{a}^{\text{m-n}}\Big]$
$=3^{6}\times2^{0}=3^{6}\times1=729$
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