Answer
$\left\{x\right\}$
Work Step by Step
Expressing both sides of the given equation, $
4^{3x}=8^{x+4}
,$ in the same base, the equation above is equivalent to
\begin{align*}
\left(2^2\right)^{3x}&=\left(2^3\right)^{x+4}
\\\\
2^{6x}&=2^{3x+12}
&(\text{use }\left(a^m\right)^n=a^{mn})
.\end{align*}
Since $a^x=a^y$ implies $x=y$, the equation above implies
\begin{align*}
6x&=3x+12
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
6x-3x&=3x-3x+12
\\
3x&=12
\\\\
\dfrac{\cancel3x}{\cancel3}&=\dfrac{12}{3}
\\\\
x&=4
.\end{align*}
Hence, the solution set to the equation $
4^{3x}=8^{x+4}
$ is $\left\{x\right\}$.
