Representing Exponents and Combinations as Summations of Smaller Combinations

Document Type : Original Article

Author

Valeo Innovations and Technology Hub in Egypt (VITE), Cairo, Egypt

Abstract

Exponential forms shall be run-time expensive, especially with the increase in the base and exponent numbers. Combinations operations shall also take excessive computing, if performed as a set of factorial multiplications based on its definition. This paper introduces with a full derivation for a useful representation of both the Exponential forms and the Combinations operations as summation of smaller combinations in the form of generic theorems. Such presentation shall be more efficient for recursive computing operations to perform expensive calculations efficiently with lower cost on parallel computing machines performing the different portions of presented summation series in parallel. The summation series components can be running concurrently on parallel computing targets. Moreover, it shall provide more insightful meanings for the exponential operations in the educational and visualization purposes. Finally, it shall provide a fair estimate for the maximum memory allocation (number of bits) needed for the obtained result of the exponential operation according to the concluded lemmas.

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