The subject of this article is exactly to analyze Beal’s conjecture and prove it. First, we classify mathematical expressions which consist of AX, BY and CZ, according to the parity of A, B and C, then get rid of two combinations of AX, BY and CZ, for they have nothing to do with the conjecture. After that, we exemplify AX+BY=CZ under the necessary constraints, where A, B, and C have at least one common prime factor. Secondly, divide AX+BY≠CZ under the necessary constraints into two kinds, and prove one kind thereof in which any two terms have a common prime factor while another term has not it. Next, under known constraints, divide another kind of AX+BY≠CZ into four inequalities. Furthermore, we derive four conclusions from the interrelation between an even number as the symmetric center and a sum of two odd numbers. This is just a preparation for proving the first two inequalities. Then, the first two inequalities are proved by the mathematical induction, fundamental theorem of arithmetic and the binomial theorem. Then again, other two inequalities are proved by the reduction to absurdity.Finally, after comparing AX+BY=CZ and AX+BY≠CZ under the necessary constraints, we came to the conclusion that Beal's conjecture is true.