We investigate the tip region of a power law fluid driven fracture propagating at constant velocity V in an impermeable linear elastic medium with the presence of fluid lag of a priori unknown length. The fracture is loaded by the internal fluid pressure pf and by the far field confining stress. The power law fluid is characterized by a fluid index n and a consistency index M. Interest in the tip region is the key to know the correct structure of the solution and to recognize the strong coupling fluid-solid which is mainly shown in a small region near the tip. The scheme is organized as follows. First, we formulate the governing equations and derive a dimensionless form of these equations which only depends on the dimensionless toughness. The nonlinear system of equations is discretized using the Gauss-Chebyshev polynomials [2,3]. This technique uses trigonometric values for the abscissas and the colocation points as made when using the Gauss-Chebyshev for solving singular integral equations corresponding to finite fractures. We transform the coordinate in order to evaluate integrals in infinite integration interval. The numerical results will include the fracture opening w, the fluid pressure pf and the relationship between the fluid lag and the toughness. We show that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behaviour in the far field is given by the solution of a semi-infinite hydraulic fracture constructed on the assumption of zero toughness solution [1]. The two asymptotes define the two limiting regimes : in the viscosity-dominated regime, the toughness of the solid is small enough that the solution can be approximated by the zero toughness solution; while in the toughness-dominated regime the fluid can be assumed to be inviscid. The intermediate part of the solution between the two asymptotes is obtained numerically.