Is there such integers $x,y$ which they're not perfect squares and they're not equal, such that:
$\sqrt{x}^\sqrt{y}$ is actually an integer? Or rational number?
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asked Sep 25, 2013 at 18:55
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$\begingroup$ I added the tag of number theory to the question. I do not know whether elementary number theory tag is more proper. $\endgroup$– ArashSep 25, 2013 at 19:09
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2 Answers
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By the Gelfond-Schneider Theorem, the number is always transcendental. In particular, it cannot be rational.
answered Sep 25, 2013 at 19:05
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By Gelfond-Schneider theorem, you can show that $\sqrt{x}^\sqrt{y}$ is irrational.