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Given Irrational Numbers a,b, Is a^b Rational?


Date: 09/26/2001 at 13:53:30
From: JM
Subject: Irrational numbers

Is it possible to demonstrate that there are irrational numbers a, b, 
where, when calculating a^b, the result is a rational number?

I've proved it is not impossible because, for example, e^(lna) = a, 
and if a*Q then lna*/Q. But I think there must be a systematic 
demonstration.

Thank you.


Date: 09/26/2001 at 14:09:50
From: Doctor Paul
Subject: Re: Irrational numbers

Consider the following:

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

Proof:

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

Otherwise, 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/   
    
Associated Topics:
High School Number Theory

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