To solve this problem, we need to determine the value of x, which represents the unknown variable in the given problem.
We are given that $g(16) = X$ and we need to find $[g(4)]^3$.
From the information given, we know that $g(f(x)) = x^3$ for all $x \ge 1$.
Let's substitute $4$ for $x$ in the equation $g(f(x)) = x^3$:
$g(f(4)) = 4^3$
$g(X) = 64$
We are also given that $[g(4)]^3 = 16$.
Let's substitute $4$ for $x$ in the equation $[g(x)]^3 = 16$:
$[g(4)]^3 = 16$
$16^3 = 16$
$4096 = 16$
Since $4096$ is not equal to $16$, we can conclude that there is an error in the problem or the information given is inconsistent.
Therefore, we cannot determine the value of the unknown variable x.
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) = X then compute $[g(4)]^3. The answer is 16. What is the value of unknown variable X?

### Answer:
