Is $2^{\sqrt{2}} +3^{\sqrt{3}}$ rational?

Question

In an exercise found in Lang's A Complete Course in Calculus, it's mentioned that it is unknown if $2^{\sqrt{2}} +3^{\sqrt{3}}$ is rational. Has this problem been solved since then?

Answer 1

We know that both $2^{\sqrt2}$ and $3^{\sqrt3}$ are transcendental $($see the Gelfond-Schneider theorem for more information$)$, but proving that their sum $($or any other non-trivial linear combination of the two$)$ is also transcendental lies beyond current knowledge. The same goes for $e+\pi$ and others.