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For all integers $n$ and $q$, prove that if the square root of $n$ is rational then the square root of $n^q$ is rational.

Having some troubles figuring out the proper logic behind this.

mzp
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  • Note: $((a)^{b})^c = a^{(bc)}$ is only guaranteed to be true under certain circumstances, one of which is when $a$ is non-negative real. Otherwise weird things can happen like $-1=1$. For a complete and thorough proof of your claim, you should note what happens if $n$ is negative and why this doesn't matter to your specific claim. – JMoravitz Oct 30 '16 at 23:48
  • This result holds even when the first number is a rational number. – Jacob Wakem Oct 31 '16 at 00:34

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Hint: $\sqrt{n^q} = \left(\sqrt{n}\right)^q$

Edit: as JMoravitz noted, this implicitly uses the fact that $\sqrt{n}$ is assumed to rational, which restricts the values to $n\geq 0$.

Michael Biro
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    You should include a disclaimer about the domain of the identity you claim. With $n=-1$ and $q=2$ this is false. – JMoravitz Oct 30 '16 at 23:51
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If the square root of $n$ is rational then there exists $a,b \in \mathbb N$ such that $$ n^{1/2} = \frac{a}{b} \Rightarrow n^{q/2} = \frac{a^q}{b^q} .$$

Then, since $a^q,b^q \in \mathbb N$, it follows that $n^{q/2}$ is rational.

mzp
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Rephrasing, if n is a ratio of perfect squares n^q would be a ratio of perfect squares. The proof follows trivially.

Jacob Wakem
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  • This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Brevan Ellefsen Oct 31 '16 at 03:12
  • @BrevanEllefsen The rephrasing is where I do the work. It does provide an adequate proof to be understood. – Jacob Wakem Oct 31 '16 at 03:18
  • Fair, which is why I didn't advocate deleting the answer (as someone suggested be done by putting the question in the queue) and instead commented. I can see your reasoning, but both since the OP asks for "proper logic" I would simply expand your answer a bit to be a bit more formal. Moreover, I wouldn't append "This is obvious." to the end of any post ever... if it were obvious the OP would not be asking :) regardless, I understand that the automatic comment might seem harsh... I was rushing through the queue and it seemed the closed pre-generated response to what I had to say.... – Brevan Ellefsen Oct 31 '16 at 03:23
  • ... but rest assured that I recommended the answer not be deleted :) – Brevan Ellefsen Oct 31 '16 at 03:23
  • @BrevanEllefsen Oh I get that a lot. My dry style gets read into in a way I don't intend. I meant that the statement is obvious AFTER you rephrase it! This is part of my (sketch) solution, not a boast. Not that this is an error on your part. – Jacob Wakem Oct 31 '16 at 03:28
  • Perfectly understandable :) Good observation nevertheless! – Brevan Ellefsen Oct 31 '16 at 03:32