early 15c., dentifricie, "substance used in cleaning the teeth," from Latin dentifricium "powder for rubbing the teeth," from dens (genitive dentis) "tooth" (from PIE root *dent- "tooth") + fricare "to rub" (see friction).
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit danta, Greek odontos (gen.), Latin dens, Lithuanian dantis, Old Irish det, Welsh dent, Old English toð, German Zahn, Gothic tunþus "tooth."
1560s, "a chafing, rubbing," from French friction (16c.) and directly from Latin frictionem (nominative frictio) "a rubbing, rubbing down," noun of action from past-participle stem of fricare "to rub, rub down," which is of uncertain origin. Watkins suggests possibly from PIE root *bhreie- "to rub, break." De Vaan suggests a PIE bhriH-o- "to cut" and compares Sanskrit bhrinanti, Old Church Slavonic briti "to shave." Sense of "resistance to motion" is from 1722; figurative sense of "disagreement, clash, lack of harmony, mutual irritation" first recorded 1761. Related: Frictional.
<a href="https://www.etymonline.com/word/dentifrice">Etymology of dentifrice by etymonline</a>
Harper, D. (n.d.). Etymology of dentifrice. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/dentifrice