Let's prove that there are two irrational numbers, call them \(a\) and \(b\), such that \(a^b\) is a rational number! And let's do it in a tweet.

the variable a raised to the power of b

Twitter proof

Let \(a = b = \sqrt 2\). If \(a^b\) is rational, then we are done; if not, take \(a = \sqrt 2^{\sqrt 2}\) and \(b = \sqrt 2\). Notice how \(a^b = (\sqrt2 ^ \sqrt2)^{\sqrt 2} = (\sqrt 2)^2 = 2\) which is obviously rational. So we have shown such \(a\) and \(b\) exist, even though we don't know if we should take \(a = b = \sqrt 2\) or \(a = \sqrt 2^{\sqrt 2}, b = \sqrt 2\).

And the actual tweet with the proof:

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