As an adjective, exact means completely correct and without any error or difference, like an exact answer or exact copy. As a verb, it means to demand and get something, often with force or authority.
It comes from Latin 'exactus', past participle of 'exigere', meaning 'to drive out, demand, examine', from 'ex-' (out) and 'agere' (to drive, do). The idea of 'driving out' the precise amount led to the sense of strict accuracy.
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