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Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$.

So, my thought process was that I could show that $2^{n} > n$ using induction, but I'm not sure how that helps to solve the original problem. Showing $2^{n} > n$ means I could take the $n^{th}$ root of each side, giving me $2 > \sqrt[n]{n}$, but that's not quite showing that it's irrational.

My other thought was proof by contradiction, I could start assuming that it's rational -- ie that there exists some integers $r$ and $s$ such that $\sqrt[n]{n} = r/s$. Then I could say that $n = r^{n} / s^{n}$. But I have no way of showing that this is impossible for any $n \geq 2$.

Any hints as to what a good direction to start in would be appreciated.

Katlyn Edwards
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2 Answers2

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Suppose there is a rational number $a/b$ such that gcd$(a,b)=1$ and $a^n/b^n=n.$ Then $a^n=b^nn.$ So $a^n$ divides $b^nn$. Since gcd$(a,b)=1 \implies $ gcd$(a^n,b^n)=1$, we know $a^n$ must divide $n$. Thus, $n \geq a^n$. But note that the integer $a$ must be at least $2$ since $(1/b)^n$ clearly cannot be $n$. Combining with the hint gives a contradiction.

StrangerLoop
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  • Wait, I'm slightly confused. How did you decide that $a$ must be at least 2? Sure, $(1/b)^{n}$ cannot be $n$, that's totally okay, but I'm unsure where you're pulling the fact that one of them must be equal to $n$. – Katlyn Edwards Jul 07 '14 at 00:29
  • I am doing a proof by contradiction--I assume that $n^{1/n}$ IS rational, and show this implies a contradiction. So, by assumption $a$ and $b$ are positive integers, and $a^n/b^n=n$. This implies $a$ cannot be $1$. So it is at least $2$. I did not understand what you mean by "one of them must be equal to $n$"; could you please clarify? I didn't say $a$ or $b$ must be $n$. – StrangerLoop Jul 07 '14 at 00:43
  • Ohhh shoot, I see, I misunderstood what you were doing in the last step then. I see. Thanks a bunch! – Katlyn Edwards Jul 07 '14 at 00:48
  • You're welcome :) – StrangerLoop Jul 07 '14 at 00:49
  • Note the proof is a special case of the proof of the Rational Root Test, whose general proof is not much more difficult. Better to invoke the more general result. – Bill Dubuque Aug 21 '23 at 18:47
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If $$\sqrt[n]{n}=\frac{a}{b}$$then $nb^n=a^n$ and if $a$ and $b$ are in lowest terms this implies that $b=1$. So it must be an integer, now its easy to show this cannot happen.

Rene Schipperus
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