Simple Criteria for \(\sqrt[n]{x}\) (n \(\in\) N, n \(\ge\) 2, x \(\in\) R) Being a Rational or an Irrational Number

Abstract

This paper presents a strong generalization of Euclid’s famous result related to \(\sqrt{2}\) being an irrational number. In particular, based on the unique prime factorization of integer numbers we obtain very simple criteria which allow us to derive necessary and sufficient conditions for \(\sqrt[n]{x}\) (n \(\in\) N, n \(\ge\) 2, x \(\in\) R) being rational or irrational. In summary, the results presented cover the complete range of cases of interest, i.e. solutions are elaborated, which – for any real number x – allow one to answer the challenging question: for which values of n, (n \(\in\) N, n \(\ge\) 2 the root \(\sqrt[n]{x}\) is still a rational number?