word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."
In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
"root up, root out," 1530s, usually figurative, from Latin extirpatus/exstirpatus, past participle of extirpare/exstirpare "root out, eradicate, pull up by the roots" (see extirpation). Related: Extirpated; extirpating; extirpable.
<a href="https://www.etymonline.com/word/inextirpable">Etymology of inextirpable by etymonline</a>
Harper, D. (n.d.). Etymology of inextirpable. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/inextirpable