Question:
For how many real values of $x$ is $\sqrt{63-\sqrt{x}}$ an integer?

Answer:
Suppose that  $k = \sqrt{63 - \sqrt{x}}$ is an integer. Then $0\le k \le \sqrt{63}$. 7 is the largest integer less than $\sqrt{63}$, and since $k$ is an integer, we have $0\le k \le 7$. Thus there are 8 possible integer values of $k$. For each such $k$, the corresponding value of $x$ is $\left(63 - k^2\right)^2$. Because  $\left(63 -
k^2\right)^2$ is positive and decreasing for $0\le k \le 7$, the $\boxed{8}$ values of $x$ are distinct.