Answer
$x=-\frac{2}{3}$
Work Step by Step
We rewrite both sides, so that the base is $2$ on both sides.
$32^{2x}=16^{x-1}$
$(2^5)^{2x}=(2^4)^{x-1}$
By the law of exponents: $(a^x)^y=a^{xy}$
Left side: $(2^5)^{2x}=2^{10x}$
Right side: $(2^4)^{x-1}=2^{4x-4}$
The rewritten equation:
$2^{10x}=2^{4x-4}$
Then, since both sides have the same base, 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 $x=y$
$10x=4x-4
\\10x-4x=-4$
$6x=-4
\\\frac{6x}{6}=\frac{-4}{6}$
$x=-\frac{2}{3}$
