Question

If $\frac{4^5+4^5+4^5+4^5}{2^5+2^5}\times \frac{6^5+6^5+6^5+6^5+6^5+6^5}{3^5+3^5+3^5}=8^n$, then n is:

Answer 1

$\frac{4^5+4^5+4^5+4^5}{2^5+2^5}\times \frac{6^5+6^5+6^5+6^5+6^5+6^5}{3^5+3^5+3^5}=8^n\frac{4^1{.4}^5}{2^1{.2}^5}\times \frac{6^1\times 6^5}{3^1{.3}^5}=8^n\frac{4^{5+1}}{2^{5+1}}\times \frac{6^{5+1}}{3^{5+1}}=8^n(\because a^n.a^m=a^{n+m})\frac{4^6}{2^6}\times \frac{6^6}{3^6}=8^n(\frac{4}{2}{)}^6\times (\frac{6}{3}{)}^6=8^n(\because (\frac{a}{b}{)}^n=\frac{a^n}{b^n})2^6\times 2^6=8^n{2}^{6+6}=8^n{2}^{12}=8^n=2^{3n}(\because (a^n{)}^m=a^{n\times m})\Rightarrow 12=3n\Rightarrow n=4$