Question

Let n ∈ N. Prove that n^(1/k) is irrational unless n = m^k

for some m ∈ N.

Answer 1

Solution. Saying that an integer k is not the n th power of an integer is equivalent to saying that the equation n^M=k has no integer solutions. Another way to say this is that the polynomial m^k-n has no integer root. Lemma therefore implies that any root of m^k-n is irrational. But k^1/m is by definition a root of this polynomial so it is irrational