Question

Find the value of $x$ for which $(\frac{3}{4}{)}^6\times (\frac{16}{9}{)}^5=(\frac{4}{3}{)}^{x+2}$

Answer 1

Given equation $(\frac{3}{4}{)}^6\times (\frac{16}{9}{)}^5=(\frac{4}{3}{)}^{x+2}$

$\Rightarrow (\frac{3}{4}{)}^6\times ((\frac{4}{3}{)}^2{)}^5=(\frac{4}{3}{)}^{x+2}$ [since $\frac{a^m}{b^m}=(\frac{a}{b}{)}^m$]
$\Rightarrow (\frac{4}{3}{)}^{x+2}=(\frac{4}{3}{)}^{-6}(\frac{4}{3}{)}^{10}$ [Since, $(a^m{)}^n=a^{m\times n}$ and $(\frac{a}{b}{)}^m=(\frac{b}{a}{)}^{-m}$

$\Rightarrow (\frac{4}{3}{)}^{x+2}=(\frac{4}{3}{)}^4$ [Since $a^m\times a^n=a^{m+n}$]
Equating powers as their bases are equal

$\therefore x+2=4⟹x=4-2$
Hence, the value of $x=2$