Answer
$x=-2$
Work Step by Step
The given equation is
$\Rightarrow 9^{2x}=3^{x-6}$
Rewrite $9$ as $3^2$.
$\Rightarrow (3^2)^{2x}=3^{x-6}$
Use $(a^n)^m=a^{n\cdot m}$
$\Rightarrow 3^{2\cdot 2x}=3^{x-6}$
Simplify.
$\Rightarrow 3^{4x}=3^{x-6}$
Equate the exponents.
$\Rightarrow 4x=x-6$
Subtract $x$ from each side.
$\Rightarrow 4x-x=x-6-x$
Simplify.
$\Rightarrow 3x=-6$
Divide each side by $3$
$\Rightarrow \frac{3x}{3}=-\frac{6}{3}$
Simplify.
$\Rightarrow x=-2$
Check: $(x=-2)$
$\Rightarrow 9^{2(-2)}=3^{-2-6}$
$\Rightarrow 9^{-4}=3^{-8}$
$\Rightarrow (3^2)^{-4}=3^{-8}$
$\Rightarrow 3^{-8}=3^{-8}$
True.
Hence, the solution is $x=-2$.
