Example 12. Compare $\sqrt{7-4 \sqrt{3}}$ with $\sqrt{5-2 \sqrt{6}}$.
Solve $\because \sqrt{7-4 \sqrt{3}}=\sqrt{(2-\sqrt{3})^{2}}$
$$
\begin{array}{l}
=2-\sqrt{3}=\frac{1}{2+\sqrt{3}}, \\
\sqrt{5-2 \sqrt{6}}=\sqrt{(\sqrt{3}-\sqrt{2})^{2}} \\
=\sqrt{3}-\sqrt{2}=\frac{1}{\sqrt{2}+\sqrt{3}} \text {. } \\
\end{array}
$$

And $2+\sqrt{3}>\sqrt{2}+\sqrt{3}$,
$$
\begin{array}{l}
\therefore \frac{1}{2+\sqrt{3}}<\frac{1}{2+\sqrt{3}}, \\
\text { Hence } \sqrt{7-4 \sqrt{3}<\sqrt{5-2 \sqrt{6}} .}
\end{array}
$$