Example 7 Let $a>0, b>0$, and $\sqrt{a}(\sqrt{a}+2 \sqrt[3]{b})$ $=\sqrt[3]{b}(\sqrt{a}+6 \sqrt[3]{b})$. Then the value of $\frac{2 a^{4}+a^{3} b-128 a b^{2}-64 b^{3}+b^{4}}{a^{4}+2 a^{3} b-64 a b^{2}-128 b^{3}+2 b^{4}}$ is
Given: From the known, taking $a=4$, we get $b=1$.
$$
\begin{aligned}
\therefore \text { Original expression } & =\frac{2 \times 4^{4}+4^{3}-128 \times 4-64+1}{4^{4}+2 \times 4^{3}-64 \times 4-128+2} \\
& =\frac{1}{2} .
\end{aligned}
$$