This root was connected to grabbing things by hand for most of Tolkien’s life. It first appeared as ᴱ√MAPA “seize” in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. map- “seize, take” and ᴱQ. maptale “robbery” (QL/59). In the contemporaneous Gnomish Lexicon its main derivative was G. mab “hand” (GL/55). In the Gnomish Lexicon, Tolkien did redefine mab as a dual form of G. mô “hand”, but it was later restored as an independent word and survived all the way into Sindarin as S. mâb “hand”.
The root reappeared as ᴹ√MAP “lay hold of with hand, seize” in The Etymologies of the 1930s with a similar set of derivatives (Ety/MAP). √MAP was mentioned again in the late 1960s in notes on Eldarin Hands, Fingers and Numerals, with glosses like “take hold of, grasp” (VT47/7) and “take away” (VT47/20). But in these same notes Tolkien made the somewhat shocking decision to abandon this root, proposing instead that S. mâb was derived from CE ✶makwā “handful” (VT47/6).
Neo-Eldarin: For purposes of Neo-Eldarin, I would ignore this very late idea, since √MAP appears so regularly in Tolkien’s earlier writings.
This root meant “fix, fasten” for much of Tolkien’s life. The earliest appearance of this root was as ᴱ√TAKA “fix, fasten, stick” in the Qenya Lexicon of the 1910s, with derivatives in that document as well as in the contemporaneous Gnomish Lexicon like ᴱQ. taka-/G. tag- “fix, make firm” and ᴱQ. tanka/G. tanc “firm, steady” (QL/88; GL/68). It also had the extended root ᴱ√TAQA “fashion” = TAK + W̯Ǝ with derivative like ᴱQ. tāqa/G. tôb(a) “shape” (QL/89; GL/71). The root ᴱ√tak “stick, [stick] in, fix, firm” was mentioned in the Early Qenya Phonology of the 1920s with numerous etymological variants like ᴱ√stak, ᴱ√taku̯, and ᴱ√tu̯ak (PE14/66).
In The Etymologies of the 1930s Tolkien gave the root ᴹ√TAK “fix, make fast” with derivatives like ᴹQ. tak-/N. taetha- “fasten”, ᴹQ. tanka/N. tanc “firm” and ᴹQ. tankil/N. tachol “pin, brooch” (Ety/TAK). The root also appeared in the first version of the Tengwesta Qenderinwa (TQ1) from the 1930s with the glosses “fix” (PE18/50) and “make” (PE18/54). √TAK reappeared in the second version of Tengwesta Qenderinwa (TQ2) from around 1950 with the gloss “fasten, fix” (PE18/100), and again (unglossed) in the Outline of Phonology (OP2) from the early 1950s as part of the derivation of ✶tankla > Q. tañkal “brooch” (PE19/83). Derivatives of this root appeared regularly in Tolkien’s later writings, making this a particularly stable root in Tolkien’s conception of Elvish.