jmc

We solve for $x^2$: $$\begin{array}{rl}192x^2-16& =0\\ 192x^2& =16\\ x^2& =\frac{1}{12}.\end{array}$$ Therefore, $x^4=\frac{1}{144}$ and $\frac{1}{x^4}=\boxed{144}$.