In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities −Δu+V(x)u+λu=|u|2∗−2u+μ|u|q−2uinRN,N≥3having prescribed mass ∫RNu2dx=a2,where μ,a>0, q∈(2,2∗), 2∗=2NN−2 is the critical Sobolev exponent, V is an external potential vanishing at infinity, and the parameter λ∈R appears as a Lagrange multiplier. Under some mild assumptions on V, combining the Pohožaev manifold, constrained minimization arguments and some analytical skills, we get the existence of normalized solutions for the problem with q∈(2,2∗). At the same time, the exponential decay property of the solutions is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for q∈2+4N,2∗).