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From the Fun Fact files, here is a Fun Fact at the Advanced level:

# Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to the power 2 and you'll get 2.

But what happens if you raise an irrational number to an irrational power? Can this ever be rational?

The answer is yes, and we'll prove it without having to find specific numbers that do the trick!

Theorem. There exist irrational numbers A and B so that AB is rational.
Proof. We know that Sqrt[2] is irrational. So, if A=Sqrt[2] and B=Sqrt[2] satisfy the conclusion of the theorem, then we are done. If they do not, then Sqrt[2]Sqrt[2] is irrational, so let A be this number. Then, letting B=Sqrt[2], it is easy to verify that AB=2 which is rational and hence would satisfy the conclusion of the theorem. QED.

This proof is non-constructive because it (amazingly) doesn't actually tell us whether Sqrt[2]Sqrt[2] is rational or irrational!

The Math Behind the Fact:
Actually, Sqrt[2]Sqrt[2] can be shown to be irrational, using something called the Gelfond-Schneider Theorem (1934), which says that if A and B are roots of polynomials, and A is not 0 or 1 and B is irrational, then AB must be irrational (in fact, transcendental).

But you don't need Gelfond-Schneider to construct an explicit example, assuming you know transcendental numbers exist (numbers that are not roots of non-zero polynomials with integer coefficients). Let x be any transcendental and q be any positive rational. Then xlog_x(q)=q so all we have to show is that log_x(q) is irrational. If log_x(q)=a/b then q=xa/b, implying that xa-qb=0, contradicting the transcendentality of x.

Su, Francis E., et al. "Rational Irrational Power." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    non-constructive proof, number theory, real analysis
Subjects:    calculus, analysis, number theory
Fun Fact suggested by:   Ravi Vakil
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