This is the 1372nd most frequent Russian word.
республике
"Республике" means "republic" (dative or prepositional case in Russian).
The word 'республике' is used in the prepositional case to denote location within a republic, following the preposition 'в'.
Я живу в республике, где природа удивительно красива.
I live in a republic where nature is wonderfully beautiful.
The word 'республике' is again in the prepositional case, referring to the setting where events occur, following the preposition 'в'.
В нашей республике проходят многочисленные фестивали.
Various festivals take place in our republic.
Here, 'республике' is used to point to the context or scope of changes occurring within the republic, again following the preposition 'в'.