"Exact a price" versus "extract a price"

Solution 1:

You are right, "exact a price" is correct. "Extract a price" is obviously a mistake.

Solution 2:

Here is the Ngram chart for "exact a price" (blue line) versus "extract a price" (red line) versus "exacts a price" (green line) versus "extracts a price" (yellow line) versus "exacted a price" (light blue line) versus "extracted a price" (maroon line) for the period 1740–2018:

A couple of things are evident from this chart. First, the "exact" forms are more common than their "extract" counterparts. Second a not inconsiderable number of people use the "extract" forms.

Besides being more popular than the "extract" form, the "exact" form is also much older. The earliest instance that a Google Books search fishes out of the ancient waters is from William Sclater, An Exposition with Notes Vpon the First Epistle to the Thessalonians (1619):

Firstly, there be that annexe consideration of Places and Countries ; which affoords no small varietie of price and worth, in things thus exchanged. To these all or some, Iustice commutatiue requires the price to be proportioned ; so that if the Seller exact a price notably exceeding the worth of the thing ; or the Buyer giue a price scienter, farre inferiour to the due value, both are said in this kind to ouer-reach. This is the generall rule to be obserued in matters of contract.

The earliest instance that Google Books finds of a similar expression using a member of the "extract" family is from "Brighton–South London," in The Railway Times (June 2, 1860):

To obtain a new [railway] line, therefore, no matter where or on what estimates, is of essential consequence to its [Chatham's] existence. Any process by which further capital might be raised could not fail to be acceptable, and therefore the indignation of the vendor of the West End line must not be exploded in Parliament; he must not be permitted to appear as a witness, with a view to detail the exact circumstances and representations under which he succeeded in extracting a price so munificent from the Brighton.

Perhaps the most telling bit of evidence in the competition between exact and extract is its lack of controversy. I consulted a number of books by usage commentators and was struck by their overwhelming tendency to ignore the issue altogether. Several books discuss extract versus extricate, and several others devote far too much time and space to the merits and demerits of "the exact same"—but only one bothers to cover exact versus extract. From Paul Brians, Common Errors in English Usage (2003):

extracting revenge/exacting revenge The use of a rare sense of "exact" ("obtain forcibly") confuses people, but the traditional phrase is "exacting revenge"; not the seemingly logical "extracting revenge."

Merriam-Webster's Eleventh Collegiate Dictionary (2003) succinctly explains how exact as a verb came by that particular meaning:

exact vt {ME, to require as payment, fr. L exactus, pp. of exigere to drive out, demand, measure, fr. ex- + agere to drive — more at AGENT} (1564) 1 : to call for forcibly or urgently and obtain {from them has been exacted the ultimate sacrifice —D.D. Eisenhower} 2 : to call for as necessary or desirable

So, according to Merriam-Webster, exact began in Middle English with the meaning "to demand as payment"—which makes the phrase "exact a price" sound quite faithful to the traditional sense of the verb. A ruthless bargainer may indeed "extract a price" from a hesitant but desperate seller under duress—but in the less extortionate sense of simply imposing a price, "exact a price" seems less tendentious. (On the other hand, I can imagine certain horizons where the two terms merge: If your book of laws imposes the price of a tooth for a tooth, do you exact the price or extract it?)