The title is pretty much clear, but here is a more precise formulation:
Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real.
I used a CAS to solve the problem and it says that the solution is $(a=0\land b>0)\lor \left(c_1\in \mathbb{Z}\land a<0\land b=c_1\right)\lor a>0$ But i think the correct answer is $(a=0\land b>0)\lor \left(c_1\in \mathbb{Q}\land \nu_2(c_1)\ge0\land a<0\land b=c_1\right)\lor a>0$ where $\nu_p$ is the $p$-adic valuation because the computer says, for example, that $(-1)^{\frac{1}{3}}=\cos(60)+\sin(60)i$.