Since $g(16) = 16,$ we have $f(g(16)) = f(16) = 16^2 = 256.$
But $f(g(16)) = f(16) = 16^3,$ so $16^2 = 16^3.$  Hence, $16^{3 - 2} = \boxed{16}.$The answer is: 16
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$  If $g(16) = 16,$ then compute $[g(4)]^3.$

### Answer:
