Etymology
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exact (adj.)
"precise, rigorous, accurate," 1530s, from Latin exactus "precise, accurate, highly finished," past-participle adjective from exigere "demand, require, enforce," literally "to drive or force out," also "to finish, measure," from ex "out" (see ex-) + agere "to set in motion, drive, drive forward; to do, perform" (from PIE root *ag- "to drive, draw out or forth, move").
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exact (v.)

"to force or compel to be paid or yielded," mid-15c., from Latin exactus, past participle of exigere "require, enforce, demand, collect (money);" see exact (adj.). Older in English than the adjective and retaining the literal sense of the Latin source. Related: Exacted; exacting.

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exacting (adj.)
"very demanding, severe in requirement," 1580s, present-participle adjective from exact (v.).
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exactness (n.)
1560s, "perfection," from exact (adj.) + -ness. Meaning "precision" is 1640s.
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exacta (n.)
type of horse-racing bet involving picking the first two horses in a race in order of finish, 1964, said to have originated in New York; from exact (adj.).
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inexact (adj.)
1791, from in- (1) "not, opposite of" + exact (adj.). Perhaps modeled on French inexact (18c.). Related: Inexactly.
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exactly (adv.)

"in an exact manner, with minute correctness," 1530s, from exact (adj.) + -ly (2). Elliptical use for "quite right" not recorded before 1869.

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exactitude (n.)

"quality of being exact, accuracy, particularity," 1734, from French exactitude (17c.), from exact, from Latin exactus (see exact (adj.)).

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exaction (n.)

late 14c., exaccioun, "action of demanding payment; imposition, requisitioning" of taxes, etc., from Old French exaccion and directly from Latin exactionem (nominative exactio) "a driving out; supervision; exaction; a tax, tribute, impost," noun of action from past-participle stem of exigere (see exact (adj.)). Meaning "a tax, tribute, toll, fee," etc. is from mid-15c.

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*ag- 

Proto-Indo-European root meaning "to drive, draw out or forth, move."

It forms all or part of: act; action; active; actor; actual; actuary; actuate; agency; agenda; agent; agile; agitation; agony; ambagious; ambassador; ambiguous; anagogical; antagonize; apagoge; assay; Auriga; auto-da-fe; axiom; cache; castigate; coagulate; cogent; cogitation; counteract; demagogue; embassy; epact; essay; exact; exacta; examine; exigency; exiguous; fumigation; glucagon; hypnagogic; interact; intransigent; isagoge; litigate; litigation; mitigate; mystagogue; navigate; objurgate; pedagogue; plutogogue; prodigal; protagonist; purge; react; redact; retroactive; squat; strategy; synagogue; transact; transaction; variegate.

It is the hypothetical source of/evidence for its existence is provided by: Greek agein "to lead, guide, drive, carry off," agon "assembly, contest in the games," agōgos "leader," axios "worth, worthy, weighing as much;" Sanskrit ajati "drives," ajirah "moving, active;" Latin actus "a doing; a driving, impulse, a setting in motion; a part in a play;" agere "to set in motion, drive, drive forward," hence "to do, perform," agilis "nimble, quick;" Old Norse aka "to drive;" Middle Irish ag "battle."

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