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Am I allowed to assume that pi is transcendental?

Any transcendental number disproves this.

I'm guessing the point is to prove the existence transcendental numbers.

Or rather, to do the proof without assuming they exist.

Oh wait! I got an idea, how about \[
\sqrt[\sqrt{2}]{2}
\]

This is obviously an irrational number

I don't see how that is relevant.

Which thing?

Without the for sight of transcendental numbers, how would you disprove it?

We know the inverse of \(y\) is also rational, let's say \(z = 1/y\)

\[ \Large
x = \sqrt{2}^{ \sqrt{2}z}
\]

\[ \Large
x = 2^{\sqrt{2}z/2}
\]

\[ \Large
x^2 = 2^{\sqrt{2}z}
\]

Now if only I can show that \[ \Large
2^{\sqrt{2}z}
\]is not rational....

'proof by contradiction'