**Authors:** Kamal Barghout

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stems from that any solution to the Beal’s equation += represents an identity equation and that the LHS of the equation represents the sum of two monomials of like terms with the value of their variables and their coefficients for each term of the equation combine by the power rules. Since the two monomials have like terms, we can factor out a common factor and add the coefficients to produce a product of two terms that can be combined by the power rules to yield the RHS of the equation. By representing a number in exponential form of single power as having a unique base-unit that repeats to comprise the number, it will be proven that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same base-unit by virtue of the two numbers having a “block-form” with a building block of their common base-unit. By conversion of the addition process of the two exponential numbers to multiplication, the GCF of the two terms on the LHS of any solution of Beal’s conjecture equation can be factored out, and by the power rules, it must be combined with the sum of the two coefficients of the two terms to yield the term on the RHS of the equation, confirming the proposition that they must have a common and distinct base-unit to successfully combine and build a single term based on the identity solution of Beal’s conjecture equation. Therefore, the process of adding the two numbers of exponential forms together on the LHS of Beal’s solution is equivalent to increasing the “size” of one of them by the other by the same number of base-units of the other to produce the number on the RHS of Beal’s solution

**Comments:** 37 Pages. The material in this article is copyrighted. Please obtain authorization from the author before the use of any part of the manuscript

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