In this paper, we study asymptotic behaviour of solutions of the following higher order nonlinear dynamic equations $$S_n^{Delta}(t,x)+{delta}p(t)f(x(g(t)))=0$$ and $$S_n^{Delta}(t,x)+{delta}p(t)f(x(h(t)))=0$$ on an arbitrary time scale $mathbb{T}$ with sup $mathbb{T}={infty}$, where n is a positive integer, ${delta}=1$ or -1 and $$S_k(t,x)={array x(t),;if;k=0,\a_k(t)S_{{kappa}-1}^{Delta}(t),;if;1{leq}k{leq}n-1,\a_n(t)[S_{{kappa}-1}^{Delta}(t)]^{alpha},;if;k=n,$$ with ${alpha}$ being a quotient of two odd positive integers and every $a_k$ ($1{leq}k{leq}n$) being positive rd-continuous function. We obtain some sufficient conditions for the equivalence of the oscillation of the above equations.