Answer
$x=\frac{1}{5}$
Work Step by Step
We rewrite both sides, so that the base is $3$ on both sides.
$27^{4x}=9^{x+1}$
$(3^3)^{4x}=(3^2)^{x+1}$
By the law of exponents:
$(a^x)^y=a^{xy}$
Left side: $(3^3)^{4x}=3^{12x}$
Right side: $(3^2)^{x+1}=3^{2x+2}$
The rewritten equation:
$3^{12x}=3^{2x+2}$
Then since the base of both sides are the same, we set the exponents equal, because the two sides are equal only if the powers are equal too: if $b^m=b^n$ and $b\ne0$, $b\ne1$ then $m=n$.
$12x=2x+2
\\12x-2x=2$
$10x=2
\\\dfrac{10x}{10}=\dfrac{2}{10}$
$x=\frac{1}{5}$
