Given that $x - \frac{1}{x} = 4$, what is $x^3 - \frac{1}{x^3}$?
One way to do that is to cube both sides of the given equation, since that would give me $x^3$ and $\frac{1}{x^3}$ terms. I get $(x - \frac{1}{x})^3 = 4^3$. Expanding the left side using the binomial formula, I get $x^3 - 3x + \frac{3}{x} - \frac{1}{x^3} = 64$. Rearranging the terms, I get $x^3 - \frac{1}{x^3} = 64 + 3x - \frac{3}{x}$. Therefore, the expression $x^3 - \frac{1}{x^3}$ has the same value for both values of $x$, and that value is 76. 