Fundamental Properties of Generalized n-th Roots of Real Numbers

In this paper we generalize our recent results related to the question “For which x \(\in\) R and n \(\in\) N, n \(\ge\) 2, \(\sqrt[n]{x}\) \(\in\) Q holds?”. We now address the more general question “For which x \(\in\) R and r \(\in\) Q, xr \(\in\) Q holds?”. We choose a step-wise approach to answer this generalized question starting with x representing an irrational number, followed by x representing a negative real number. Finally, we comprehensively answer the question for x representing a natural number, which also directly leads to solutions covering the practically relevant case of x representing a positive rational number. One important goal of our research has been to improve the understanding of the properties of irrational numbers as well as simplifying existing solutions for n-th root problems. To demonstrate the potential usage of our results related to fundamental properties of n-th roots of real numbers, by way of example, we indicate how our results allow us to obtain elegant solutions for several rather challenging tasks related to irrational \(\sqrt[n]{x}\).