Find all solutions to the equation $\displaystyle\sqrt[3]{3 - x} = -\frac{3}{2}$.
To solve for $x$, I need to get rid of the cube root on the left side. I can do that by cubing both sides of the equation, since $(a^3)^{\frac{1}{3}} = a$ for any number $a$. Cubing the left side gives me $3 - x$, and cubing the right side gives me $-\frac{27}{8}$. So I have $3 - x = -\frac{27}{8}$. That gives me $-x = -\frac{51}{8}$. That gives me $x = \frac{51}{8}$. I get $\displaystyle\sqrt[3]{3 - \frac{51}{8}} = \sqrt[3]{-\frac{27}{8}} = -\frac{3}{2}$, which is true. 