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Given $a,d,c,d\in\mathbb{N}$, define

$$ X = a^\frac{1}{b} $$ $$ Y = c^\frac{1}{d} $$ $$ Z = X-Y $$

I'm pretty sure the following is true:

For any $X\neq Y$ that satisfy the above relationships, $Z$ is rational if and only if $X$ and $Y$ are rational.

I've been wanting to prove this relationship out of the interesting fact that it would show that if $X$ and $Y$ are irrational, then $\{X\}$ and $\{Y\}$ are unique, where $\{x\}$ represents the fractional part of a number, $x$. Or am I mistaken? Using the binomial theorem looks to be a good starting point, as seen here in an earlier post, but I've tried going from there and it seems to get murky, because you get alternating sequences of radicals. Any thoughts would be greatly appreciated!

Edit: The result of this question seems to show that this is in fact true. Am I mistaken?

Samuel
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    You need to make your statement more precise, because it is false as stated now : consider for example $a=3,b=2,c=9,d=4$. – Ewan Delanoy Feb 27 '18 at 16:05
  • Thanks for pointing that out, I will update. – Samuel Feb 27 '18 at 16:06
  • "For any X≠Y, Z is rational if and only if X and Y are rational" Hmmm... Try $X=1+\pi$ and $Y=\pi$. – Did Feb 27 '18 at 16:26
  • $\pi$ cannot be represented as an integer raised to a unit fraction... – Samuel Feb 27 '18 at 16:28
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    Yeah, we are slowly moving towards a sensible statement... – Did Feb 27 '18 at 16:44
  • Sorry, I'm a hobbyist, not a professional. With every question, I learn something new. I need to remember to be a precise as possible so I leave no holes. Thanks for the feedback! – Samuel Feb 27 '18 at 16:49
  • With "suitable" modifications (i.e., not as it is now), your statement is correct. However, I still do not see how the answer follows from the other thread. – Paolo Leonetti Feb 27 '18 at 18:00
  • I'm an optimist... :) @Paolo Leonetti: could you please make suitable modifications in an answer and show with such modifications it is true? That would be great! – Samuel Feb 27 '18 at 18:04
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    I am not posting this as an answer because it is a classical question. See this MSE question and reference therein: https://math.stackexchange.com/questions/150141/linear-independence-of-roots-over-q and also this article of Iure Boreico at p.87: http://www.math.harvard.edu/hcmr/issues/2.pdf – Paolo Leonetti Feb 27 '18 at 18:05
  • Thanks so much! – Samuel Feb 27 '18 at 18:06
  • You're welcome. Have a good reading :) – Paolo Leonetti Feb 27 '18 at 18:07

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