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<p>Indeed, 1/2*2 = 1 is the desired counterexample, and I think the strongest version of the statement you’re thinking of is that the product of a rational and an irrational is irrational (identical proof to Quelloquialism’s, slightly stronger statement). </p>
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<p>I think you might be thinking of something like an <em>integer</em> raised to a “funky power” is not an integer in the first place – not sure if there’s anything useful like this. It’s in fact characteristic of non-integers, rather than integers, to become integers when raised to funky powers – i.e., take any strange root of an integer, and you have a billion examples.</p>