Simple Tests for n-th Roots of Natural Numbers being Natural Numbers and Elementary Methods to Determine Their Values

In this paper we tackle the challenging problem to determine, in a simple but reliable way, whether – for a given, arbitrary number x, x ≥ 2 – the n-th root of x produces a rational or an irrational result, i.e. we determine whether n√x ∈ Q or n√x ∉ Q . To solve this problem in a straightforward manner we make use of the prime factorization of x. As a main contribution we present a generally applicable algorithm to decide whether n√x ∈ Q (for n,x ∈ N\{1} ) and if so, to determine the resulting value. Moreover, we design several tests which can be applied to determine, for which values of n n√x ∈, Q if the natural number x satisfies a given set of properties. Quite often the tests proposed will allow us to answer the question “ n√x ∈ Q ?” in a matter of seconds. Finally, we demonstrate that, for a very high percentage of all natural numbers x, x ≥ 2, it is impossible to find even a single n ∈N,n≥2 such that n√x ∈ Q.