Answer
$x=5$
Work Step by Step
The given expression is
$\Rightarrow (27)^{2x+2}=81^{x+4}$
Use $27=3^3$ and $81=3^4$.
$\Rightarrow (3^3)^{2x+2}=(3^4)^{x+4}$
Use $(a^n)^m=a^{nm}$.
$\Rightarrow (3)^{3(2x+2)}=(3)^{4(x+4)}$
Use distributive property.
$\Rightarrow (3)^{6x+6}=(3)^{4x+16}$
Equate the exponents.
$\Rightarrow 6x+6=4x+16$
Add $-4x-6$ to each side.
$\Rightarrow 6x+6-4x-6=4x+16-4x-6$
Simplify.
$\Rightarrow 2x=10$
Divide each side by $2$.
$\Rightarrow \frac{2x}{2}=\frac{10}{2}$
Simplify.
$\Rightarrow x=5$
