Radical comes straight from the Late Latin adjective rādicālis “having roots, rooted,” first occurring about a.d. 400 in Contra Faustum (“Against Faustus the Manichaean”) by St. Augustine of Hippo. Rādicālis is a derivative of the noun rādix (inflectional stem rādīc- ) “root (of a plant, tooth, hair), root (of a family, stock, breed), (etymological) root.” The mathematical sense “denoting the radical sign which indicates the root of a number” dates from the late 17th century. Radical in its political sense dates from the late 18th century in England and the first half of the 19th century in the United States.
Latin rādix comes from wrād-, one of the variants of the Proto-Indo-European root wrād, werād, wred- “root, branch.” From this same variant, Latin also has rāmus “branch” (the root, so to speak, of English ramify ), while Greek has rhádix (stem rhádik- ). Another variant of the root is the possible source of Greek rhiza, source of English rhizome; other variants include Aeolic briza, brisda and Mycenaean wriza. Wrād- regularly becomes wrōt- in Proto-Germanic, the ultimate source of the English word root.

Did you mean ultracritical?