Irrational Numbers x,y, x^y Rational?

Date: 09/28/2001 at 14:37:31
From: Joe Palmer
Subject: Proof: Are there any irrational numbers x and y such that x^y 
is rational?

I am taking college level Calculus I, and we were given a bonus 
question: Are there any irrational numbers X and Y such that X^Y is 
rational? 

I have tried several examples and have concluded that there are not, 
but I think our professor is looking for a law or theorum to prove 
or disprove this statement. I am currently researching the Internet 
in an attempt to find such a law, etc. Any help would be greatly 
appreciated. 

Thanks, Joe

Date: 09/28/2001 at 16:12:05
From: Doctor Paul
Subject: Re: Proof: Are there any irrational numbers x and y such that 
x^y is rational?

Hi Joe.  

Such numbers do in fact exist and I can prove it without giving you 
the actual numbers x and y. Consider the following:

Claim: There exists an irrational number r such that r^sqrt(2) is  
       rational.

Proof:

The proof has two cases:

Case 1:  sqrt(3)^sqrt(2) is a rational number

If sqrt(3)^sqrt(2) is rational, then we're done because r = sqrt(3) is 
the desired value of r.

Case 2:  sqrt(3)^sqrt(2) is not a rational number

In this case, x = sqrt(3)^sqrt(2) is irrational. Then x^sqrt(2) = 
sqrt(3)^2 = 3, which is rational.

Therefore, either sqrt(3) or sqrt(3)^sqrt(2) is an irrational number r 
such that r^sqrt(2) is rational.

I think this establishes the result you want - there exist irrational
numbers x and y such that x^y is rational.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/