Algebra 3 - by Vlad Vasilescu (W)

We can see that $(\sqrt{5+2\sqrt{6}})^x \cdot (\sqrt{5-2\sqrt{6}})^x = 1$

We say $(\sqrt{5+2\sqrt{6}})^x = t \Rightarrow (\sqrt{5-2\sqrt{6}})^x = \frac{1}{t}$

We solve for t in $t + \frac{1}{t} = 10 \Rightarrow t_{1} = 5 + 2\sqrt{6}$ and $t_{2} = 5 - 2\sqrt{6 }$

For $t = 5 + 2\sqrt{6}$ we get x=2 and for $t = 5 - 2\sqrt{6}$ we get x=-2 .

Then , our answer will be 2 .