This interesting root is זכה and the secondary form זכך. They both mean "to be clear, clean, pure." One fairly obvious related word is zakh זך - "pure, clean" as in shemen zayit zakh שמן זית זך - "pure olive oil."
A noun that I didn't realize was related is zekhukhit זכוכית - "glass". It only appears once in the Tanach (Iyov 28:17) - and while the glass back then wasn't transparent like it generally is today, it was certainly more clear than other solids.
From the literal "pure" and "clear" in Biblical Hebrew, the root took on more of a metaphorical sense in Rabbinic Hebrew. Just like we say in English that an innocent person has been "cleared" of charges and has a "clean" record, the verb זכה means "to make someone innocent", and that innocent person is zakai זכאי - "innocent."
An innocent, pure person is considered "worthy" and even "deserving" (of goodness). And so another meaning of the verb זכה is "to deserve, to attain." This leads us to the noun zekhut זכות - which according to Klein has 5 different meanings:
- privilege, benefit
- legal right, title
- favor, advantage
- merit, virtue
- credit side of an account, asset
In English there is a clear distinction between rights and privileges. In Hebrew there is discussion of the difference between zekhut and chova חובה - "obligation", but I'm not sure how you could contrast rights and privileges without using a foreign word for privilege like פריבילגיה.
From zekhut comes the word zakaut זכאות - "entitlement." But since zekhut can have different connotations, it doesn't necessarily carry the negative associations that "entitlement" can have in English today.
There are two other Hebrew roots that are likely related to זכה/זכך. One is זגג - it is the Aramaic equivalent of זכך, and is found in Hebrew as well. The zag זג is the skin of the grape (Bamidbar 6:4), and Klein says it is probably "allusion to the transparency of the skin of the grape." In Rabbinic Hebrew, a glassmaker is a zagag זגג, and today if you need to get the windows on your car replaced you go to a zagagut זגגות - "glazier(y)."
The other related root, according to Klein, is זקק. This verb means "to purify, to refine." Distilled water is mayim mezukakim מים מזוקקים, and batei zikuk בתי זיקוק are "refineries."
The other meaning of זקק - "to bind, force, compel" (as in זקוק zakuk - "in need of") is not related to this one, and neither is the root זיק meaning to "to spark, to sparkle" (as in zikukim זקוקים - "fireworks"). But I think we found plenty of words that are related, we are not zakukim for more..