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/ |
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