Example 1. If $a>1, b$ is a positive rational number, $a^{b}+a^{-0}$ $=2 \sqrt{2}$, find the value of $a^{b}-a^{-b}$.
Consider the following solution:
Let $a^{b}=x$, then $x+\frac{1}{x}=2 \sqrt{2}$.
Transform it into $x^{2}-2 \sqrt{2} x+1=0$.
Solving yields $x_{1}=\sqrt{2}+1, x_{2}=\sqrt{2}-1$.
$$
\text { When } \begin{aligned}
x & =\sqrt{2}+1, \\
& a^{b}-a^{-b} \\
& =\sqrt{2}+1-\frac{1}{\sqrt{2}+1} \\
& =\sqrt{2}+1-(\sqrt{2}-1) \\
& =2
\end{aligned}
$$

When $x=\sqrt{2}-1$,
$$
\begin{aligned}
& a^{b}-a^{-b} \\
= & \sqrt{2}-1-\frac{1}{\sqrt{2}-1} \\
= & \sqrt{2}-1-(\sqrt{2}+1) \\
= & -2 .
\end{aligned}
$$

This solution is incorrect. The error lies in neglecting the conditions $a>1$ and $b$ being a positive integer - if these conditions were considered, it would be clear that $a^{b}>1$. Therefore, the correct answer should be 2.