Is there a “simple” mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?

**Answer**

Here’s a cute and lovely theorem.

There exist two irrational numbers $x,y$ such that $x^y$ is rational.

*Proof.* If $x=y=\sqrt2$ is an example, then we are done; otherwise $\sqrt2^{\sqrt2}$ is irrational, in which case taking $x=\sqrt2^{\sqrt2}$ and $y=\sqrt2$ gives us: $$\left(\sqrt2^{\sqrt2}\right)^{\sqrt2}=\sqrt2^{\sqrt2\sqrt2}=\sqrt2^2=2.\qquad\square$$

(Nowadays, using the Gelfond–Schneider theorem we know that $\sqrt2^{\sqrt2}$ is irrational, and in fact transcendental. But the above proof, of course, doesn’t care for that.)

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