Free Space Optics is a field of communication with atmosphere as the channel, imposing a number of challenges in transmission of information. This paper presents a novel analytical approach to analyze the performance of multi-branch EGC combiner operating over independent fading environments in Free Space Optical System by implementing two-point Padé approximants theory in a moment-based approach. The Log-normal, Nakagami-n (Rician), and Rayleigh fading channels, representing weak-to-strong turbulence regimes, are considered for investigation. By deriving the exact closed-form expressions of moments of the output signal-to-noise ratio and joint probability density function of the sum of N-independent fading random variables, the significant performance metrics, such as the amount of fading, average output signal-to-noise ratio, and spectral efficiency are evaluated. Moreover, the average symbol error rate, outage probability and channel capacity of the system are assessed by approximating moment generating function using two-point Padé Approximants theory. Various numerical results achieved from proposed theoretical approach are validated for accuracy using computer simulations.