I think the answer to the question in the title is "yes", because $9^{2/3}$ is irrational by an argument similar to the accepted answer in this question. Or am I mistaken?
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4Yes. Or $x = 2$, $q = 1/4$ – Simon S Dec 01 '14 at 00:00
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It looks like you answered your own question. Yes, your answer is correct – Ben Grossmann Dec 01 '14 at 00:00
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Actually, it'd be quite interesting to know how 'big' is the set of irrationals with this form. – hjhjhj57 Dec 01 '14 at 01:24
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... Yep. For example, $x = 2$, $q = \frac{1}{4}$.
$2^{2(1/4)} = 2^{1/2}$, an irrational number.
Robert Soupe
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Bliebervik
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Yes, it can be, x=1. But, it is not necessarily rational, as Simon S. pointed out, x=2, q=1/4.