The work has a recognizable structure, for the most part indicated by the author himself, which can be summarized as follows:The Prince starts by describing the subject matter it will handle. In the first sentence Machiavelli uses the word "state" (Italian stato which could also mean "status") in order to neutrally cover "all forms of organization of supreme political power, whether republican or princely". The way in which the word state came to acquire this modern type of meaning during the Renaissance has been the subject of many academic discussions, with this sentence and similar ones in the works of Machiavelli being considered particularly important.
Machiavelli said that The Prince would be about princedoms, mentioning that he has written about republics elsewhere (possibly referring to the Discourses on Livy although this is debated), but in fact he mixes discussion of republics into this in many places, effectively treating republics as a type of princedom also, and one with many strengths. More importantly, and less traditionally, he distinguishes new princedoms from hereditary established princedoms. He deals with hereditary princedoms quickly in Chapter 2, saying that they are much easier to rule. For such a prince, "unless extraordinary vices cause him to be hated, it is reasonable to expect that his subjects will be naturally well disposed towards him". Gilbert (1938:19â23), comparing to traditional presentations of advice for princes, stated that the novelty in chapters 1 and 2 is the "deliberate purpose of dealing with a new ruler who will need to establish himself in defiance of custom". Normally, these types of works were addressed only to hereditary princes. He thinks Machiavelli may have been influenced by Tacitus as well as his own experience, but finds no clear predecessor for this.
This categorization of regime types is also "un-Aristotelian" and apparently simpler than the traditional one found for example in Aristotle's Politics, which divides regimes into those ruled by a single monarch, an oligarchy, or by the people, in a democracy. He also ignores the classical distinctions between the good and corrupt forms, for example between monarchy and tyranny.
Xenophon, on the other hand, made exactly the same distinction between types of rulers in the beginning of his Education of Cyrus where he says that, concerning the knowledge of how to rule human beings, Cyrus the Great, his exemplary prince, was very different "from all other kings, both those who have inherited their thrones from their fathers and those who have gained their crowns by their own efforts".
Machiavelli divides the subject of new states into two types, "mixed" cases and purely new states.

Question: How are republics treated in this work?
Answer: Reublics are effectively treated as a type of princedom.
Alternative answer: as a type of princedom