Scientific computation relies heavily on root finding numerical methods, which allow accurate modelling of complicated systems in physics, engineering, and other fields, resulting in important scientific discoveries. To solve nonlinear equations, this work presents an eighth-order hybrid iterative method that combines a two-step fourth-order strategy with the second-order Newton-Raphson method. With an efficiency rating of 1.5157, this three-step approach evaluates five functions (two functions and three first-order derivatives) and produces results better than some existing iterative methods. Numerical comparisons with proposed method are presented using MAPLE software, and its relevance to real-world models such as computing force between particles and solving Van der Waals equation for volume of a real gas is illustrated. The proposed method is equally suitable for solving both scalar and vector forms of nonlinear equations.