We present a precision study of the $psiptopi^0 J/psi$ and $eta J/psi$ decay modes. The measurements are obtained using $106times10^6$ $psi$ events accumulated with the BESIII detector at the BEPCII $ee$ collider operating at a center-of-mass energy corresponding to the $psip$ mass. We obtain $mathcal{B}(psiptopi^0 J/psi)=(1.26pm0.02{rm (stat.)}pm0.03{rm (syst.)})times 10^{-3}$ and $mathcal{B}(psiptoeta J/psi)=(33.75pm0.17{rm (stat.)}pm0.86{rm (syst.)})times 10^{-3}$. The branching fraction ratio $R=frac{mathcal{B}(psiptopi^0 J/psi)}{mathcal{B}(psiptoeta J/psi)}$ is determined to be $(3.74pm0.06 {rm(stat.)}pm0.04 {rm(syst.)})times 10^{-2}$. The precision of these measurements of $mathcal{B}(psiptopi^{0} J/psi)$ and $R$ represent a significant improvement over previously published values.
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