Etymology
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*ak- 
Proto-Indo-European root meaning "be sharp, rise (out) to a point, pierce."

It forms all or part of: acacia; acanthus; accipiter; acer; acerbic; acerbity; acervate; acervulus; acescent; acetic; acid; acicular; acme; acne; acrid; acridity; acrimony; acro-; acrobat; acromegaly; acronym; acrophobia; acropolis; acrostic; acrylic; acuity; aculeate; acumen; acupressure; acupuncture; acute; aglet; ague; Akron; anoxic; awn; coelacanth; dioxin; deoxy-; eager; ear (n.2) "grain part of corn;" edge (n.); egg (v.) "to goad on, incite;" eglantine; epoxy; ester; exacerbation; hammer; hypoxia; mediocre; oxalic; oxide; oxy-; oxygen; oxymoron; paragon; pyracanth; paroxysm; selvage; vinegar.

It is the hypothetical source of/evidence for its existence is provided by: Greek akros "at the end, at the top, outermost; consummate, excellent," akis "sharp point," akros "at the farthest point, highest, outermost," akantha "thorn," akme "summit, edge," oxys "sharp, bitter;" Sanskrit acri- "corner, edge," acani- "point of an arrow," asrih "edge;" Oscan akrid (ablative singular) "sharply;" Latin acer (fem. acris) "sharp to the senses, pungent, bitter, eager, fierce," acutus "sharp, pointed," acuere "to sharpen," acerbus "harsh, bitter," acere "be sharp, be bitter," acus "a needle, pin," ocris "jagged mountain;" Lithuanian ašmuo "sharpness," akstis "sharp stick;" Old Lithuanian aštras, Lithuanian aštrus "sharp;" Old Church Slavonic ostru, Russian óstryj "sharp;" Old Irish er "high;" Welsh ochr "edge, corner, border;" Old Norse eggja "goad;" Old English ecg "sword;" German Eck "corner."
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*bhau- 
Proto-Indo-European root meaning "to strike."

It forms all or part of: abut; baste (v.3) "beat with a stick, thrash;" battledore; beat; beetle (n.2) "heavy wooden mallet;" botony; boutonniere; butt (n.1) "thick end;" butt (v.) "strike with the head;" buttocks; button; buttress; confute; halibut; rebut; refute; sackbut; turbot.

It is the hypothetical source of/evidence for its existence is provided by: Latin *futare "to beat" (in compounds); Old English beadu "battle," beatan "to beat," bytl "hammer, mallet."
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*kae-id- 

*kaə-id-, Proto-Indo-European root meaning "to strike."

It forms all or part of: abscise; avicide; biocide; caesarian; caesura; cement; chisel; -cide; circumcise; circumcision; concise; decide; decision; deicide; excise (v.); excision; felicide; feticide; filicide; floricide; fratricide; fungicide; gallinicide; genocide; germicide; herbicide; homicide; incise; incision; incisor; infanticide; insecticide; legicide; liberticide; libricide; matricide; parricide; patricide; pesticide; precise; precision; prolicide; scissors; senicide; spermicide; suicide; uxoricide; verbicide.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit skhidati "beats, tears;" Latin caedere "to strike down, fell, slay;" Lithuanian kaišti "shave;" Armenian xait'em "to stab;" Albanian qeth "to shave;" Middle Dutch heien "to drive piles," Old High German heia "wooden hammer," German heien "beat."

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*mele- 

*melə-, Proto-Indo-European root meaning "to crush, grind," with derivatives referring to ground or crumbling substances and crushing or grinding instruments.

It forms all or part of: amyl; amyloid; blintz; emmer; emolument; immolate; maelstrom; mall; malleable; malleolus; mallet; malleus; maul; meal (n.2) "edible ground grain;" mill (n.1) "building fitted to grind grain;" millet; mola; molar (n.); mold (n.3) "loose earth;" molder; ormolu; pall-mall.

It is the hypothetical source of/evidence for its existence is provided by: Hittite mallanzi "they grind;" Armenian malem "I crush, bruise;" Greek mylos "millstone," myle "mill;" Latin molere "to grind," mola "millstone, mill," milium "millet;" Old English melu "meal, flour;" Albanian miel "meal, flour;" Old Church Slavonic meljo, Lithuanian malu, malti "to grind;" Old Church Slavonic mlatu, Russian molotu "hammer."

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