Answer
$x=3$.
Work Step by Step
$\left(\frac{1}{6}\right)^{2x}=216^{1-x}$
Rewriting $(\frac{1}{6})^{2x}$ as $6^{-2x}$ and $216$ as $6^{3}$, we have
$6^{-2x}=(6^{3})^{(1-x)}$
$\implies 6^{-2x}=6^{(3-3x)}$
$\implies -2x=3-3x$
$\implies -2x+3x=3$ or $x=3$.
Check:
$\left(\frac{1}{6}\right)^{2(3)}=6^{-6}$
$216^{1-3}=(6^3)^{-2}=6^{-6}$
