Question
Evaluate:
$\frac{15^{4}\times18^{3}}{3^{3}\times5^{2}\times12^{2}}$
$\frac{15^{4}\times18^{3}}{3^{3}\times5^{2}\times12^{2}}$
✓
Answer
We have, $\frac{15^{4}\times18^{3}}{3^{3}\times5^{2}\times12^{2}}=\frac{(3\times5)^{4}\times(2\times3^{2})^{3}}{3^{3}\times5^{2}(2^{2}\times3)^{2}}$
$=\frac{3^{4}\times5^{4}\times2^{3}\times3^{6}}{3^{3}\times5^{2}\times2^{4}\times3^{2}}$ $\Big[\because 18=2\times3\times3\text{ and }12=2\times2\times3\Big]$
$=\frac{3^{4+6-3-2}\times5^{4-2}}{2^{4-3}}$ $\Big[\because(\text{a}\times\text{b})^{ \text{m}}=\text{a}^{\text{m}}\times\text{b}^{\text{m}}\Big]$
$=\frac{3^{5}\times5^{2}}{2}=\frac{243\times25}{2}=\frac{6075}{2}$ $\Big[\because\frac{\text{a}^{\text{m}}}{\text{a}^{\text{n}}}=\text{a}^{\text{m-n}}\text{ and }\text{a}^{\text{m}}\times\text{a}^{\text{n}}=\text{a}^{\text{m+n}}\Big]$
$=\frac{3^{4}\times5^{4}\times2^{3}\times3^{6}}{3^{3}\times5^{2}\times2^{4}\times3^{2}}$ $\Big[\because 18=2\times3\times3\text{ and }12=2\times2\times3\Big]$
$=\frac{3^{4+6-3-2}\times5^{4-2}}{2^{4-3}}$ $\Big[\because(\text{a}\times\text{b})^{ \text{m}}=\text{a}^{\text{m}}\times\text{b}^{\text{m}}\Big]$
$=\frac{3^{5}\times5^{2}}{2}=\frac{243\times25}{2}=\frac{6075}{2}$ $\Big[\because\frac{\text{a}^{\text{m}}}{\text{a}^{\text{n}}}=\text{a}^{\text{m-n}}\text{ and }\text{a}^{\text{m}}\times\text{a}^{\text{n}}=\text{a}^{\text{m+n}}\Big]$
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