It may not surprise you that price and prize are cognates, but two other common words pertaining to value, and additional words derived from them, share their common ancestor.
Price and prize stem from the Latin noun pretium, meaning “prize” or “reward,” or “value” or “worth.” Price is both a verb meaning “set value” and a noun meaning “value” (including a reference to something challenging or unpleasant that is nevertheless necessary as a condition of achieving a desired result). The adjective pricey means “expensive,” while priceless means not “without a price” but “valued beyond price” and refers to an object or an experience worth so much to a person or people in general because of aesthetic considerations or an emotional attachment that no monetary value can be assigned to it.
To overprice is to place excessive value on something for sale, while to be priced out of a commodity is to be unable to afford it. A price tag is a piece of paper or other material attached to a product that indicates its monetary value. Something offered for half-price will be sold for half of its marked value. “Prix fixe,” adopted directly from French and meaning “fixed price,” describes a set meal offered by a restaurant at a specific cost, as opposed to multiple dishes available, according to a displayed menu, for various prices.
To prize is to ascribe value, and a prize is something given in recognition of an achievement. As an adjective, prize means “worthy of a prize,” and prizeworthy itself is also an adjective. A prizefight is a professional boxing match, and a participant of such an event is a prizefighter, though these terms have largely been superseded by “boxing match” and boxer.
The Latin forebear pretium was altered to precium, and being aware of this form makes it easier to connect precious, meaning “of great worth” (and often pertaining to aesthetic or emotional worth, as in ascribing value to time, rather than monetary value), to price and prize. Precious also has the sense of “esteemed,” in describing a cherished friend, though it also has a pejorative sense of “affected,” or “overly refined in manner.” The adjectival form is preciously, and the quality of being precious is preciousness.
Praise, too, is derived from pretium by way of precium. That word, meaning “celebrate” or “commend,” or as a noun “commendation” or “worship” (or, less often, “merit” or “value”), is also the root of appraise, meaning “set a value on,” and the noun form appraisal. A praiseworthy act is one that merits commendation.
To appreciate is to esteem or value, or to be conscious of, and an act of doing so is one of appreciation; the adjective appreciable means “able to be measured or perceived.” Depreciate, by contrast, means “lower in esteem or value”; in a taxation context, it refers to deducting a portion of the original cost of something as its value decreases with age and use. The verb deprecate, meaning “belittle,” “play down,” or “disapprove of,” is unrelated. (Its root word pertains to prayer; originally, to deprecate was to avert something undesired by praying.)