$4 \cdot 51$ Find the sum of all roots of the following equation
$$
\sqrt[4]{x}=\frac{12}{7-\sqrt[4]{x}} .
$$
[Solution] Let $y=\sqrt[4]{x}$, then $y=\frac{12}{7-y}$, which means $y^{2}-7 y+12=0$,

Solving for $y$,
$$
\begin{array}{ll}
\text { we get } & y_{1}=3, y_{2}=4 . \\
\text { Therefore, } & x_{1}=3^{4}, x_{2}=4^{4} .
\end{array}
$$

Thus, the sum of all roots $x_{1}+x_{2}=3^{4}+4^{4}=337$.