In the paper two commutativity theorems are proved: (i) If R is a semi prime ring and $n>1$ a fixed positive integer such that either $[[x,y]^n-[x^n,y^n],x]=0$ or $[(x\circ y)^n-(x^n\circ y^n),x]=0$, then $R$ is commutative (ii) If $R$ is a ring in which for each $x,y$ in $R$ there exists a positive integer $n=n(x,y)>1$ such that $(xy)^n=yx$, then $R$ is commutative.