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

1. ### rangerjoe

1
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.

2. ### HallsofIvy

40,528
Staff Emeritus
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.

3. ### matticus

107
if p is prime and divides m^n, p must divide m...