Answer
The solution is $x=-\dfrac{b}{a}$
Work Step by Step
$a^3x^3+b^3=0\hspace{0.7cm} \color{blue}{\text{Given equation}}$
$\Rightarrow a^3x^3=-b^3$
$\Rightarrow x^3=-\dfrac{b^3}{a^3}$
$\Rightarrow x=-\bigg(\dfrac{b^3}{a^3}\bigg)^{1/3}=-\dfrac{(b^3)^{1/3}}{(a^3)^{1/3}}=-\dfrac{b}{a}$
The solution is $x=-\dfrac{b}{a}$