A steady laminar flow of an incompressible non‐Newtonian fluid (inelastic power law fluid,n<1) in the inlet region of rectangular ducts was studied by the theoretical analysis using finite difference methods and the experiment on pressure drops. Theoretical predictions were compared with experimental data and analytical results available and the agreement was shown to be satisfactory. Therefore, the axial pressure distributions and the velocity profiles in the inlet region were discussed on the basis of typical computations. It was shown that the velocity in the duct center and the pressure drop of a power law fluid were smaller than those of a Newtonian fluid; the inlet length of a power law fluid was larger than that of a Newtonian fluid and increased with the decreasing power law index. Also, the additional pressure loss in the inlet region was smaller in a power law fluid than in a Newtonian fluid and decreased with the decreasing power law index. In addition, when the effect of duct shape was considered, it was found that the velocity in the duct center and the pressure drop of a power law fluid decreased with the increasing aspect ratio of the duct, as in the case of a Newtonian fluid (they are at a maximum in the square duct).