Can you prove that there is at least one instance where an irrational number raised to an irrational power...

arithmetic/irrational.power New

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Problem

Can you prove that there is at least one instance where an irrational
number raised to an irrational power yields a rational number?

The proof is rather amusing.

-- Aaron M Morris <ammorris@tuba.aix.calpoly.edu>

Show solution

You mean, besides exp[log[2]]?

-- Dude Neergaard <neergaar@schaefer.math.wisc.edu>

Maybe he wants a proof that doesn't use the knowledge that e and log[2]
are irrational.

-- Mark van Hoeij <hoeij@sci.kun.nl>

How about:
There is a well-known proof that sqrt(2) is irrational (look it up in
any maths book).  Now consider x = sqrt(2)^sqrt(2).  It's either
rational or irrational.  If the former, then we're done.  Otherwise,
consider x^sqrt(2) which is another irrational number raised to an
irrational power.  But x^sqrt(2) = sqrt(2)^(sqrt(2)*sqrt(2)) =
sqrt(2)^2 = 2 which is certainly rational.
So, in one of the cases, we have an irrational^irrational = rational.

-- ard@siva.bris.ac.uk (University of Bristol Physics Department)