Take $$n>2 \in \mathbb{N}$$. Then $$2^{1/n}$$ is irrational. Suppose $$2^{1/n} = a/b$$. Then $$2b^n=a^n$$, so $$b^n+b^n=a^n$$ giving a counter-example to Fermat's Last Theorem.

Thus every $$n^{th}$$ root of 2 is irrational from 3 onwards.