Example 16. Solve the equation: $(\sqrt{5+2 \sqrt{6}})^{x}$
$$
+(\sqrt{5-2 \sqrt{6}})^{x}=10 .
$$
Solve $\because y^{\prime} 5-2 \sqrt{6}=\frac{1}{\gamma^{i} 5+2 v^{\prime} 6}$,
$\therefore$ the equation can be rewritten as $(\sqrt{5+2 \sqrt{6}})^{x}$ $+\frac{1}{(\sqrt{5+2 \sqrt{6}})^{x}}=10$.
Let $\mathrm{y}=(\sqrt{5+2 \sqrt{6}})^{\mathrm{x}}$, then $\mathrm{y}+\frac{1}{y}=10$,
which simplifies to $y^{2}-10 y+1=0$,
solving this, we get $y=5 \pm 2 \sqrt{6}$.
This gives $5+2 \sqrt{6}=(\sqrt{5+2 \sqrt{6}})^{x}$,

yielding $x_{1}=2$;
and $5-2 \sqrt{6}=(\sqrt{5+2 \sqrt{6}})^{x}$,

yielding $x_{2}=-2$.
Upon verification, $\mathrm{x}_{1}=2$ and $\mathrm{x}_{2}=-2$ are both roots of the original equation.