Expert: Sherry Wallin - 9/9/2010

Question
QUESTION: How to prove that an irrational number raised the power of an irrational number results in a rational number.

ANSWER: Suraj~

If it were true maybe then you could prove it. pi^pi ~= 36.46215961...
Try others, like sqrt2^sqrt2 doesn't work either.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: I am sorry. What I wanted to know was that if it is possible to prove that irrational number raised to an irrational power results in an rational number. From your answer it seems that an irrational number raised to an irrational number is always an irrational number. It is obviously not true, as the square root of 2 raised to itself has been proved to be a rational number.

ANSWER: Suraj~

I answered your original question as if you wanted to prove that any (every, all) irrational numbers raised to an irrational number is rational and all you have to do is find one case where it isn't true. Now you are saying how do you prove that there exists an irrational number that when raised to itself then it is rational and you have proved it by just providing an existing case. It still is not clear what it is you really want to prove.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Sir, I am sorry, the language of my question was not clear. What I want to ask if it is possible that in all cases, an irrational number raised to an irrational power always yields a rational number. You have raised pi to the power of pi. But can we prove that its result is necessarily an irrational number?  

Answer
Suraj~

No, not all irrational numbers raised to irrational powers will be rational. In the case of pi raised to the pi power you could try a proof by contradiction to show that the result is irrational but pi is a transcendental number and not algebraic, and thus would yield another irrational number.

I am not sure what level of math you are studying so you might google transcendental numbers and algebraic as well.

Math Prof