Answer
$-1$.
Work Step by Step
${{3}^{2x}}=\frac{1}{9}$
Simplify the exponent ${{3}^{2x}}=\frac{1}{9}$ as follows.
$\begin{align}
& {{3}^{2x}}=\frac{1}{9} \\
& {{3}^{2x}}={{\left( \frac{1}{3} \right)}^{2}} \\
& {{3}^{2x}}={{3}^{-2}}
\end{align}$
Use principle of exponent equality and solve:
$\begin{align}
& 2x=-2 \\
& x=-1
\end{align}$
Check:
Substitute $x=-1$ in the given equation.
$\begin{matrix}
{{3}^{2\left( -1 \right)}}\overset{?}{\mathop{=}}\,\frac{1}{9} \\
\text{ }{{3}^{-2}}\overset{?}{\mathop{=}}\,\frac{1}{9} \\
\text{ }\frac{1}{{{3}^{2}}}\overset{?}{\mathop{=}}\,\frac{1}{9} \\
\text{ }\frac{1}{9}=\frac{1}{9} \\
\end{matrix}$
Thus, the obtained solution is correct.
