Literally, to drive from within the limits or borders. Hence,

To destroy utterly; to drive away; to extirpate; as, to exterminate a colony, a tribe or a nation; to exterminate inhabitants or a race of men.

To eradicate; to root out; to extirpate; as, to exterminate error, heresy, infidelity or atheism; to exterminate vice.

To root out, as plants; to extirpate; as, to exterminate weeds.

In algebra, to take away; as, to exterminate surds or unknown quantities.