jmc

A real number $x$ is in the domain of $f(x)$ unless $x^2=7$, $x^3=8$, or $x^4=9$.

The solutions to $x^2=7$ are $x=\sqrt{7}$ and $x=-\sqrt{7}$, which sum to $0$.

The only solution to $x^3=8$ is $x=2$.

The solutions to $x^4=9$ are $x=\sqrt[4]{9}$ and $x=-\sqrt[4]{9}$, which sum to $0$.

Thus, the sum of all $x$ not in the domain of $f$ is $0+2+0=\boxed{2}$.