Answer
$x = -\dfrac{1}{6}$
Work Step by Step
Note that $\dfrac{1}{\sqrt[3]{3}} = \dfrac{1}{3^{\frac{1}{3}}}=3^{-\frac{1}{3}}$.
Thus, the given expression is equivalent to:
$9^x=3^{-\frac{1}{3}}$
Write $9$ as $3^2$ to obtain:
$(3^2)^x=3^{-\frac{1}{3}}$
Use the rule $(a^m)^n=a^{mn}$ to obtain:
$3^{2x} = 3^{-\frac{1}{3}}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$2x = -\dfrac{1}{3}$
Divide 2 on both sides of the equation to obtain:
$x = -\dfrac{1}{6}$