The hydrodynamics of free rising buoyant solid spheres under sub-critical, critical, and super-critical conditions in a non-Newtonian pseudoplastic liquid was investigated for the first time ever. The main parameters studied were the drag coefficient, terminal velocity, and trajectory of rise. From the results obtained in this study and also those previously published, it was found that for Reynolds numbers between 135 and 7000, the drag coefficient of the rising spheres was constant at a value close to 0.95. In the Reynolds range 7000-20 000, as the Reynolds number increased, the drag coefficient slightly decreased to 0.7. It was observed that all spheres in this sub-critical Reynolds range 135-20 000 displayed a spiraling trajectory. In the Reynolds range 20 000-55 000, the drag coefficient decreased substantially to a value of 0.2. At Reynolds numbers greater than 55 000, it appeared the solid spheres rose under super-critical conditions. The drag coefficient was constant at a value close to 0.2 and the sphere trajectory was very close to linear. In addition, the first drag correlation for rising solid spheres in non-Newtonian power-law liquids for 0 < Re-1 < 25 000 is proposed.