<s>[INST] exponential equation $$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$
So I have squared both sides and got:
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$
I don't know what to do now
 [/INST] Observe that $(5 - 2\sqrt{6}) = \frac{1}{5+2\sqrt{6}}$.
So, if we set $a = 5+ 2 \sqrt{6}$, we have
\begin{equation}
a^x +a^{-x} +2 = 100.
\end{equation}
The above expression is symmetric in $x$, so if $x = k$ is a solution, $x = -k$ is also a solution. Consider $f(x) = a^x +a^{-x} +2$ for $x>0$. You can show that this is an increasing function for $x > 0$. This implies that there can be at most one value of $x>0$ for which the equality is satisfied. You can check by substitution that $x = 2$ is a solution. So, the only solutions for the equation are $x = +2$ and $x=-2$.
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