Here's a breakdown:
* Sanskrit: "avaram" (formal singular)
* Prakrit: "ava" (formal singular, evolved from "avaram")
* Hindi: "aap" (formal singular and plural, evolved from "ava")
Over time, "ava" transformed into "aap" and also expanded its usage to include the formal plural. This signifies the respect and formality inherent in the word "aap" when addressing someone.
So, while there isn't a single direct root, "avaram" serves as the etymological origin of the word "aap" in Hindi.