Can I cut off the third prong of a plug to make it fit into a 2 prong socket?

NO - This is NOT safe.

If you have a 3-pronged tool, but an outlet that only takes 2 prong devices, you cannot use the two together. It is NOT safe to cut the extra prong off of your tool's plug; the third prong is the grounding prong. If you cut it off or use a plug adapter that is not grounded, you will disable a feature that is designed to protect against electrical shock.