13 is one of the worst numbers in the world!!

$\begin{aligned} \left(\sqrt{2}\right)^x + \left(\sqrt{3}\right)^x & = \left(\sqrt{13}\right)^{\frac{x}{2}} \\ 2^{\frac{x}{2}} + 3^{\frac{x}{2}} & = 13^{\frac{x}{4}} \\ & = \left(4+9\right)^{\frac{x}{4}} \\ \Rightarrow 2^{\frac{x}{2}} + 3^{\frac{x}{2}} & = \left(2^2+3^2 \right)^{\frac{x}{4}} \\ \text{For } x = 4 \quad \Rightarrow 2^{\frac{4}{2}} + 3^{\frac{4}{2}} & = \left(2^2+3^2 \right)^{\frac{4}{4}} \\ 2^2+3^2 & = 2^2+3^2 \end{aligned}$

$\Rightarrow x = \boxed{4}$