Example. Simplify $\frac{2 \sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$.
$$
\begin{array}{c}
\text { Solve the original expression }=\frac{(\sqrt{2}+\sqrt{3})^{2}-(\sqrt{5})^{2}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} \\
=\frac{(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{3}-\sqrt{5})}{\sqrt{2}+\sqrt{3}+\sqrt{5}} . \\
=\sqrt{2}+\sqrt{3}-\sqrt{5} .
\end{array}
$$

This method is very effective, but as a math problem, this alone cannot achieve the desired effect.

Solving math problems using the idea of combining numbers and shapes often gives people a fresh and extraordinary feeling.

Construct $\triangle A B C$ such that $a=\sqrt{2}, b=\sqrt{3}, c=$ $\sqrt{5}$,
let $p=$
$$
\frac{a+b+c}{2} \text {. }
$$

It is easy to see that $\triangle A B C$ is a right triangle.

Construct $\odot O$ inscribed in the right triangle $\triangle A B C$, with radius $r$, and note that $r=\frac{a+b-c}{2}=$
$$
\begin{array}{l}
\sqrt{2}+\frac{\sqrt{3}-v^{\prime}}{2} . \\
\because S_{\triangle A B C}=\frac{a b}{2}=\frac{\sqrt{2} \times \sqrt{3}}{2}=\frac{\sqrt{6}}{2}, \\
\therefore 2 \sqrt{6}=4 S_{\triangle A B C} . \\
\text { Also, } S_{\triangle A B C}=r p, \\
\therefore \text { the original expression }=\frac{4 S_{\triangle A B C}}{2 p}=\frac{4 r p}{2 p}=2 r \\
=\sqrt{2}+\sqrt{3}-\sqrt{5} .
\end{array}
$$

From this, it can be seen that combining numbers and shapes is a powerful tool for broadening thinking, a 'higher level of thinking'.
