The Nth Root of N

Date: 11/28/2000 at 20:25:09
From: Gavin King
Subject: Nth roots

Dear Dr. Math,

Is the nth root of n (a whole number other than 1) ever a rational 
number?

Thanks, Gavin

Date: 11/28/2000 at 20:32:50
From: Doctor Schwa
Subject: Re: Nth roots

No. It definitely can't be rational. If you want a proof, try the same 
proof that you have probably seen for the square root of 2 being 
irrational.

If (a/b)^n = n, and a/b is in lowest terms, then a^n = n*b^n, so a is 
divisible by n, ... or wait, it isn't, quite. For example, if n were 4 
then a would only have to be divisible by 2 to make the left side 
divisible by 4.

Okay, if p is a prime divisor of n, then p must be a prime divisor of 
a also, whereupon you can prove that either n is divisible by p^n 
(which it can't be, since p^n is too big) or b is divisible by p, 
whereupon a/b is not in lowest terms after all, a contradiction.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/