On the Rationality and Transcendentality of Solutions to the Equation x^n + y^n = z^n

In this paper, the author develops 2 theorems that serve as extensions to the well-known Fermat’s Last Theorem in the field of number theory. The first proposed theorem in this paper states that if there exist numbers x, y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then both/either x and/or y must be irrational. The second proposed theorem in this paper states that if there exist a number x, a transcendental number y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then x must be irrational. The proposed theorems in this paper expand on the notion that if there exist numbers x, y, and an integer z such that x^n + y^n = z^n for any integer n > 2, then at least x or y is not an integer, which is stated in Fermat’s Last Theorem.