In　this　paper,　we　study　the　permutation　property　of　pentanomials　with　the　form　xrh(xpm−1)　over　Fp2m,　where　p∈{2,3}.　More　precisely,　based　on　some　seventh-degree　and　eighth-degree　irreducible　pentanomials　over　F2,　we　present　eight　classes　of　permutation　pentanomials　over　F22m　by　determining　the　solutions　of　some　equations　with　low　degrees.　In　addition,　based　on　the　investigation　of　algebraic　curves　associated　with　fractional　polynomials　over　finite　fields,　eight　classes　of　permutation　pentanomials　over　F32m　are　discovered　by　choosing　some　seventh-degree　irreducible　pentanomials　over　F3.　Finally,　several　classes　of　permutation　pentanomials　and　heptanomials　over　F22m　and　F32m　are　derived　from　known　permutation　polynomials　on　μ2m+1　and　μ3m+1,　respectively,　where　μd　is　the　set　of　d-th　roots　of　unity.　©　2023　Elsevier　Inc.