1. The problem statement, all variables and given/known data "Prove m√n is not a rational number for any natural numbers with n,m > 1, where n is not an mth power" 2. Relevant equations Natural numbers for us start at 1. Since we know n is not an mth power, then n [itex]\neq[/itex] km for an arbitrary integer k. 3. The attempt at a solution I believe this would follow a similar argument for the classic proof that √2 is irrational. We set up the contradiction that m√n is actually rational. Then am=nbm for some a,b [itex]\in[/itex] [itex]Z[/itex], b [itex]\neq[/itex] 0, a,b are irreducible. Then am is a multiple of n, so a is a multiple of n. I'm assuming I need to prove this as a lemma. I've attempted proving this by modular arithmetic. Prove by contrapositive (If a is not a multiple of n, then am is not a multiple of n). But then there would be n-1 cases to check (infinitely many), so I follow with induction. Base case: n=2, a [itex]\equiv[/itex] 1 (mod 2), then a2 [itex]\equiv[/itex] 12 = 1 [itex]\equiv[/itex] 1 (mod 2). Base case checks out, so we can assume the inductive hypothesis that this is true for all n. To show show n+1, the (mod n) turns into (mod n+1), so how do I use the inductive hypothesis on this? This is the point where I get lost. Any suggestions?