Prove that if m, n are natural, then the root

[rangerjoe]

Hi,

I've encountered this exercise which I'm having a hard time proving. It goes like this:
Prove that if m and n are natural, then the nth root of m is either integer or irrational.

Any help would be greatly appreciated. Thanks.

 

[HallsofIvy]

I would imagine you would start by proving it for m= p, a prime number. That would make it easy to prove "If integer k is not divisible by p then neither is kⁿ" so you could mimic Euclid's proof that \sqrt{2} is irrational.

After that, look at products of prime.

 

[matticus]

if p is prime and divides m^n, p must divide m...