Here'a a really simple and intuitive proof that sqrt(2) is irrational, which extends immediately to the nth root of any rational number that isn't a perfect nth power. It goes like this. Let r be a rational number, and let a, b be integers such that r=a/b and (a,b)=1 (so that a/b is r in lowest terms). Then clearly r2=a2/b2, and it is clear that (a2,b2)=1, so that this is r2 in lowest terms. Thus we see that the square of any rational number, written in lowest terms, has a perfect square in both the numerator and denominator. So we can conclude that if a rational number does not have this property, it is not the square of a rational number, and so its square root is irrational. Since 2 is not a perfect square, sqrt(2) is irrational. This seems a lot more straightforward than the standard proof by contradiction. Is this well known, and why isn't it more popular?