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We obtain a general exact solution of the Einstein field equations for the anisotropic Bianchi type I universes filled with an exponential-potential scalar field and study their dynamics. It is shown, in agreement with previous studies, that for a wide range of initial conditions the late-time behavior of the models is that of a power-law inflating Friedmann-Robertson-Walker (FRW) universe. This property does not hold, in contrast, when some degree of inhomogeneity is introduced, as discussed in our following paper.