Answer
$x=6$
Work Step by Step
$\Big(\dfrac{1}{9}\Big)^{x}=27^{2-x}$
Rewrite $27$ as $3^{3}$ and $9$ as $3^{2}$:
$\Big(\dfrac{1}{3^{2}}\Big)^{x}=(3^{3})^{2-x}$
Take $3^{2}$ to the numerator by changing the sign of its exponent:
$(3^{-2})^{x}=(3^{3})^{2-x}$
Multiply the exponents on both sides of the equation:
$3^{-2x}=3^{6-3x}$
If $3^{-2x}=3^{6-3x}$, then $-2x=6-3x$:
$-2x=6-3x$
Solve for $x$:
$-2x+3x=6$
$x=6$