Answer
$\dfrac{4}{9}$
Work Step by Step
Note that:
$$27=3^3$$ and $$81=3^4$$
Thus, the given equation is equivalent to:
$$\left(3^3\right)^{3x}=3^4$$
Use the rule $\left(a^m\right)^n=a^{mn}$ to obtain:
\begin{align*}
3^{3\cdot 3x}&=3^4\\
3^{9x}&=3^4
\end{align*}
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
\begin{align*}
9x&=4\\\\
\frac{9x}{9}&=\frac{4}{9}\\\\
x&=\frac{4}{9}
\end{align*}
