exact

Exact derives from a form of the Latin verb exigere, meaning "to drive out, to demand, or to measure." (Another descendant of exigere is the word exigent, which can mean "demanding" or "requiring immediate attention.") Exigere, in turn, was formed by combining the prefix ex- with the verb agere, meaning "to drive." Agere has been a prolific source of words for English speakers; it is the ancestor of agent, react, mitigate, and navigate, just to name a few. Incidentally, if you are looking for a synonym of the verb exact, you could try demand, call for, claim, or require.

Do you exact or extract revenge?

The verb exact (as in, "exacting revenge" or "exacting a promise") is not as commonly encountered as the adjective exact, (as in "an exact copy" or "exact measurements"). Sometimes people will mistakenly use the more common verb extract when they really want exact. Extract can refer to removing something by pulling or cutting or to getting information from someone who does not want to give it. While both words refer to getting something they are used in different ways. You extract a tooth, but you exact revenge.