One of my favorite proofs is the following:
Claim: There exists irrational numbers $\alpha$ and $\beta$ such that $\alpha^{\beta}$ is rational.
Proof: Let $\alpha = \sqrt{2}^{\sqrt{2}}$ and $\beta = \sqrt{2}$ so $\alpha, \beta \notin \mathbb{Q}$. Therefore, $$\alpha^{\beta} = (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{{\sqrt{2}}\cdot{\sqrt{2}}} = (\sqrt{2}^{2}) = 2 $$ So $\alpha^{\beta} \in \mathbb{Q}$.
With that said: are there are any other proofs to claims, theorems, lemma's, etc. that are short and powerful like this one? Please do share. Visual proofs are also welcome!
For example, let $x$ be a root of the equation: $$\left(\sqrt2\right)^x=3.$$ Prove that this $x$ is an irrational number (it's not so hard by contradiction).