Example 16 The greatest integer not exceeding $(\sqrt{7}+\sqrt{5})^{6}$ is $\qquad$ .
Solution: Let $a=\sqrt{7}+\sqrt{5}, b=\sqrt{7}-\sqrt{5}$. Then
$$
\begin{array}{l}
a+b=2 \sqrt{7}, a b=2 . \\
a^{3}+b^{3}=(a+b)^{3}-3 a b(a+b) \\
=44 \sqrt{7}, \\
a^{6}+b^{6}=\left(a^{3}+b^{3}\right)^{2}-2(a b)^{3}=13536 . \\
\because 0<b<1, \\
\therefore 0<b^{6}<1 .
\end{array}
$$

Therefore, the integer part of $a^{6}$ is 13535.