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/
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