Answer
$\left\{-\dfrac{13}{3}\right\}$
Work Step by Step
Expressing both sides of the given equation, $
2^{3x-7}=8^{2x+2}
,$ in the same base results to
\begin{align*}\require{cancel}
2^{3x-7}&=\left(2^3\right)^{2x+2}
\\\\
2^{3x-7}&=2^{6x+6}
&(\text{use }\left(a^m\right)^n=a^{mn})
.\end{align*}
Since $b^x=b^y$ implies $x=y$, the equation above implies
\begin{align*}\require{cancel}
3x-7&=6x+6
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
-6-7&=6x-3x
\\
-13&=3x
\\\\
-\dfrac{13}{3}&=\dfrac{\cancel3x}{\cancel3}
\\\\
-\dfrac{13}{3}&=x
.\end{align*}
Hence, the solution set of the equation $
2^{3x-7}=8^{2x+2}
$ is $
\left\{-\dfrac{13}{3}\right\}
$.
