Answer
The solution set is $\left\{-\sqrt2, 0, \sqrt2\right\}$.
Work Step by Step
To solve the given equation, make the two sides have the same base.
Note that $9 = 3^2$ so the given equation is equivalent to:
$3^{x^3} = (3^2)^x$
Use the rule $(a^m)^n = a^{mn}$ to obtain:
$3^{x^3} = 3^{2x}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$x^3 = 2x$
Subtract $2x$ to both sides of the equation to obtain:
$x^3-2x=0$
Factor out the GCF $x$ to obtain:
$x(x^2-2)=0$
Equate each factor to $0$, then solve each equation to obtain:
$\begin{array}{ccc}
&x=0 &\text{ or } &x^2-2=0
\\&x=0 &\text{ or } &x^2=2\end{array}$
Take the square root of both sides of the second equation obtain:
$\begin{array}{ccc}
&x=0 &\text{ or } &x=\pm \sqrt{2}\end{array}$
Thus, the solution set is $\left\{-\sqrt2, 0, \sqrt2\right\}$.
