[bizaro86]

Originally Posted by psyang

Ok, new puzzle for the weekend.

Prove that there exists two irrational numbers, a and b, such that a^b (a to the power of b) is rational.

As a reminder, rational numbers are numbers that can be expressed as a ratio of two integers (like 1/3, or 12). Irrational numbers are numbers that aren't rational.

Spoiler!

I was working on this as a laws of exponents problem and was messing around with 'e'. I suspect there is a general proof here but one example proves that it does exist, and e to the power of ln(2) equals 2, and both e and ln(2) are irrational.

I vaguely remember proving than ln(2) is irrational, but that was probably 15 years ago, so I'm going to go with, "it can be shown that ln(2) and e are irrational" and 2 is obviously rational. Neither have terminating or repeating decimals.