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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.

Contradiction.

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

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