Abstract: In this paper, we use the theory of value distribution and study the uniqueness of meromorphic functions. We will prove the following result: Let f(z) and g(z) be two transcendental meromorphic functions, p(z) a polynomial of degree k, n≥max{11, k+1} a positive integer. If fⁿ(z)f(z) and gⁿ(z)g(z) share p(z)CM, then either f(z)=c_1e^cp(z)dz, g(z)=c₂e^cp(z)dz ,where c₁, c₂ and c are three constants satisfying (c₁c₂)ⁿ⁺¹c₂₌-1 or f(z)≡tg(z) for a constant t such that tⁿ⁺¹=1.

Key words: meromorphic function, polynomial, constant, zero point