Abstract
This paper investigates the algebraic and combinatorial structures governing the exponentiation of generalized hybrid numbers, a non-commutative ring that unifies complex, hyperbolic, and dual number systems. Hybrid rings arise naturally when extending sequences, such as Fibonacci-type or Leonardo-type numbers, into algebraic structures with multiple components. We rigorously demonstrate that every hybrid number satisfies a unique quadratic characteristic equation dictated strictly by its real trace and hybrid character. This foundational property establishes a surjective, non-injective mapping between hybrid numbers and classical second-order linear recurrence sequences, specifically defining the fundamental and fundamental-Lucas characteristic sequences. The study formalizes the anti-involutive property of the hybrid conjugate and provides comprehensive, structural proofs for the multiplicative character identity across all positive, negative, and degenerate integer powers. The core contribution of this work is the development of four distinct, mathematically unified frameworks for evaluating arbitrary integer powers of a hybrid number: a sequential recursive framework, a decoupled Lucas-equivalence formulation, an analytic structural root expansion utilizing generalized Binet formulas, and a purely real combinatorial expansion that entirely bypasses complex algebraic arithmetic. Finally, these established architectures are extended to resolve the complex non-commutative mixed products of arbitrary hybrid powers, providing exact, closed-form combinatorial identities for both positive and negative exponent combinations.