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[parent] irrational to an irrational power can be rational (Result)

Let $ A=\sqrt{2}^{\sqrt{2}}$. If $ A$ is a rational number, then it has the required property. If $ A$ is an irrational number, let $ B = A^{\sqrt{2}}$, then $ B=\sqrt{2}^2 = 2$ is a rational. Hence an irrational number to an irrational power can be a rational number. (In fact, it follows from the Gelfond-Schneider Theorem that $ A$ is transcendental, and therefore irrational.)



"irrational to an irrational power can be rational" is owned by Koro. [ full author list (3) | owner history (1) ]
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See Also: techniques in mathematical proofs


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Cross-references: transcendental, rational, irrational number, rational number
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This is version 7 of irrational to an irrational power can be rational, born on 2003-06-24, modified 2004-01-31.
Object id is 4389, canonical name is IrrationalToAnIrrationalPowerCanBeRational.
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AMS MSC: 11A99 (Number theory :: Elementary number theory :: Miscellaneous)
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