1. The problem statement, all variables and given/known data Let a and n be positive integers. Prove that a^(1/n) is either an integer or is irrational. 2. Relevant equations 3. The attempt at a solution Proof: If a^(1/n) = x/y where y divides x, then we have an integer. If a^(1/n) = x/y where y does not divide x, then a = (a^(1/n))^n = x^n/y^n is NOT an integer since y^n does not divide x^n. However, this is a contradiction as we declared a to be a positive integer. Thus, a^(1/n) must be an integer. However, is neglecting the important part of irrationality. In my proof, I have convinced myself that a^(1/n) is an integer. But this is obviously not true as 4^(1/3) is irrational. Where did I go wrong? Perhaps there is another case?