We　establish　optimal　error　bounds　on　time-splitting　methods　for　the　nonlinear　Schrodinger　equation　with　low　regularity　potential　and　typical　power-type　nonlinearity　f(rho)　=　rho(sigma),　where　rho　:=　|psi|(2)　is　the　density　with　psi　the　wave　function　and　sigma　＞　0　the　exponent　of　the　nonlinearity.　For　the　first-order　Lie-Trotter　time-splitting　method,　optimal　L-2-norm　error　bound　is　proved　for　L-infinity-potential　and　sigma　＞　0,　and　optimal　H-1-norm　error　bound　is　obtained　for　W-1,W-4-potential　and　sigma　＞=　1/2.　For　the　second-order　Strang　time-splitting　method,　optimal　L-2-norm　error　bound　is　established　for　H-2-potential　and　sigma　＞=　1,　and　optimal　H-1-norm　error　bound　is　proved　for　H-3-potential　and　sigma　＞=　3/2　(or　sigma　=　1).　Compared　to　those　error　estimates　of　time-splitting　methods　in　the　literature,　our　optimal　error　bounds　either　improve　the　convergence　rates　under　the　same　regularity　assumptions　or　significantly　relax　the　regularity　requirements　on　potential　and　nonlinearity　for　optimal　convergence　orders.　A　key　ingredient　in　our　proof　is　to　adopt　a　new　technique　called　regularity　compensation　oscillation　(RCO),　where　low　frequency　modes　are　analyzed　by　phase　cancellation,　and　high　frequency　modes　are　estimated　by　regularity　of　the　solution.　Extensive　numerical　results　are　reported　to　confirm　our　error　estimates　and　to　demonstrate　that　they　are　sharp.