Time scales have been introduced in order to unify the theories of differential and difference equations and in order to extend these cases to many other so-called dynamic equations. in this paper we consider a linear dynamic equation on a time scale together with a perturbed equation. We show that, if certain exponential dichotomy conditions are satisfied, then for any solution of the perturbed equation there exists a solution of the unperturbed equation that asymptotically differs from the solution of the perturbed equation no more than the order of the perturbation term. in order to show this perturbation theorem, we use many properties of the exponential function on time scales and derive several bounds for certain monomials that appear in the dynamic version of Taylor's formula.