Electroosmotic flow of Power-law fluids over a surface with arbitrary zeta potentials is analyzed. The governing equations including the nonlinear Poisson-Boltzmann equation, the Cauchy momentum equation and the continuity equation are solved to seek exact solutions for the electroosmotic velocity, shear stress, and dynamic viscosity distributions inside the electric double layer. Specifically, an expression for the general Smoluchowski velocity is obtained for electroosmosis of Power-law fluids in a fashion similar to the classic Smoluchowski velocity for Newtonian fluids. The existing Smoluchowski slip velocities under two special cases, (i) for Newtonian fluids with arbitrary zeta potentials and (ii) for Power-law fluids with small zeta potentials, can be recovered from our derived formula It is interesting to note that the general Smoluchowski velocity for non-Newtonian Power-law fluids is a nonlinear function of the electric field strength and surface zeta potentials, this is due to the coupling electrostatics and non-Newtonian fluid behavior, which is different from its counterpart for Newtonian fluids This general Smoluchowski velocity is of practical significance in determining the flow rates in microfluidic devices involving non-Newtonian Power-law fluids