Find the positive difference between the two solutions to the equation $\displaystyle\sqrt[3]{4 - \frac{x^2}{3}} = -2$.
To solve this equation, I need to get rid of the cube root on the left side. I can do that by cubing both sides of the equation, which gives me $4 - \frac{x^2}{3} = -8$. Now I can isolate $x^2$ by subtracting 4 from both sides and multiplying by -3, which gives me $x^2 = 36$. To find the solutions for $x$, I need to take the square root of both sides, which gives me $x = \pm 6$. The positive difference between these two solutions is the absolute value of their subtraction, which is $|6 - (-6)| = 12$. 