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Is $12^{1/3}$ irrational? Give a proof that justifies your answer

So far I have: Suppose $12^{1/3}$ is rational.This means there exists integers a and b such that $12^{1/3} = \frac{a}{b}$ where $\gcd(a,b) = 1$.

Then, $12 = \frac{a^3}{b^3}$ so $a^3 = 12b^3$. This means $12|a^3$. However since $12$ is not prime, you can't say $12|a$.

Please help!

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    $\begingroup$ You can prove it's rational by Rational Root Theorem: $x^3-12 =0 $ $\endgroup$
    – GohP.iHan
    May 2, 2015 at 1:54
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    $\begingroup$ @GohP.iHan: I think you may have meant to say that you can use the rational root theorem to prove that the root is irrational. $\endgroup$
    – Sid
    May 2, 2015 at 2:15
  • $\begingroup$ Haha yeah! Thanks! $\endgroup$
    – GohP.iHan
    May 2, 2015 at 12:03

1 Answer 1

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Note that $3|a^3$, so $3|a$. So then $3|b^3$ and $3|b$. Can you take it from here?

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