A scheme of approximation of the Kohn-Sham exchange potential v x is proposed, making use of a partitioning of v x into the long-range Slater v S and the short-range response v resp components. A model potential v resp mod has been derived from dimensional arguments. It possesses the proper short-range behavior and the atomic-shell stepped structure characteristic for v resp. When combined with the accurate v S, v resp mod provides an excellent approximation to the exchange potential of the optimized potential model v x OPM. With the generalized-gradient approximation to v S v resp mod provides an efficient density-functional-theory approach that fits closely the form of the accurate exchange potential and yields reasonably accurate exchange and total energies as well as the energy of the highest occupied orbital.