We give a combinatorial algorithm to find a maximum packing of hypertrees in a capacitated hypergraph. Based on this we extend to hypergraphs several algorithms for the k-cut problem, that are based on packing spanning trees in a graph. In particular we give a γapproximation algorithm for hypergraphs of rank γ, extending the work of Ravi and Sinha [22] for graphs. We also extend the work of Chekuri, Quanrud and Xu [7] in graphs, to give an algorithm for the k-cut problem in hypergraphs that is polynomial if k and the rank of the hypergraph are fixed. We also give a combinatorial algorithm to solve a linear programming relaxation of this problem in hypergraphs.