<s>[INST] Find the number of values of $x$ in $(\sqrt{2})^x+(\sqrt{3})^x = (\sqrt{13})^{\frac{x}{2}}$. Question:

Find the number of values of $x$ with $$(\sqrt{2})^x+(\sqrt{3})^x = (\sqrt{13})^{\frac{x}{2}}.$$

My attempt:
I tried setting $p = \sqrt{2}$, $q = \sqrt{3}$, and resetting the equation to:
$$p^x + q^x = \sqrt{p^4+q^4}^{\frac{x}{2}}$$
But I don't still think that expansion would probably solve it. Anyone has a better way ?
 [/INST] Divide by $(\sqrt{13})^{\large\frac{x}{2}}$, and write $x = 4y$. You obtain
$$\left(\frac{4}{13}\right)^y + \left(\frac{9}{13}\right)^y = 1.$$
For $0 < a < 1$, the sole solution of $a^y + (1-a)^y = 1$ is $y = 1$, since $y\mapsto a^y$ and $y\mapsto (1-a)^y$ are both strictly decreasing.
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