A Visual Tree Model and Algorithmic Implementation of the Binomial Theorem via Sequential Base-10 Carry Propagation

We present a general algorithm for computing the n^p power of any non-negative integer nfor any integer power p ≥ 2. It operates by splitting the number n into its leading part andits final digit. The powers of these parts are weighted by binomial coefficients , and theresulting cross-terms are combined with sequential base-10 carry propagation. The derivationof this algorithm occurred within the context of a mental-arithmetic shortcut that the authordeveloped on his own, prior to understanding any related known mathematical methods. Themethod is mathematically equivalent to the binomial theorem as it is applied digit-by-digit tothe number, and its two special cases, p = 2 and p = 3, coincide with the duplex and anurupyenamethods for Vedic mathematics, respectively. The provided diagrammatic structure serves asthe backbone of this method. Additionally, there is a working implementation of the methodin Python, along with the necessary discussion of the method in relationship to other existingmathematical literature.