Chapter 2

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

Abstract/Preface

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 √𝑥 𝑛 ∈ Q or √𝑥 𝑛 ∈ 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 √𝑥 𝑛 ∈ 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, √𝑥 𝑛 ∈ Q if the natural number x satisfies a given set of properties. Quite often the tests proposed will allow us to answer the question “ √𝑥 𝑛 ∈ 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 √𝑥 𝑛 ∈ Q.