Answer
$x=1$
Work Step by Step
The given equation is
$\Rightarrow 4^{3x}=8^{x+1}$
Rewrite $4$ as $2^2$ and $8=2^3$.
$\Rightarrow (2^2)^{3x}=(2^3)^{x+1}$
Use $(a^n)^m=a^{n\cdot m}$
$\Rightarrow 2^{2\cdot 3x}=2^{3\cdot (x+1)}$
Simplify.
$\Rightarrow 2^{6x}=2^{ 3x+3}$
Equate the exponents.
$\Rightarrow 6x=3x+3$
Subtract $3x$ from each side.
$\Rightarrow 6x-3x=3x+3-3x$
Simplify.
$\Rightarrow 3x=3$
Divide each side by $3$
$\Rightarrow \frac{3x}{3}=\frac{3}{3}$
Simplify.
$\Rightarrow x=1$
Check: $(x=1)$
$\Rightarrow 4^{3(1)}=8^{1+1}$
$\Rightarrow 4^{3}=8^{2}$
$\Rightarrow 64=64$
True.
Hence, the solution is $x=1$.
