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### Irrational Numbers x,y, x^y Rational?

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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/
```
Associated Topics:
College Number Theory
High School Number Theory

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