This is the 739th most frequent Croatian word.
američkih
The Croatian word "američkih" is the genitive plural form of "američki," meaning "American."
In this case, 'američkih' modifies 'Država' (in its genitive plural form) and agrees in case, number, and gender. It indicates the attribute that the States are American, being part of the context of their history.
Učili su o povijesti američkih Država.
They studied the history of the American States.
Here, 'američkih' is the genitive plural form of the adjective 'američki', describing 'knjiga' (books), also in the genitive plural. This is used because the word 'set' (set) determines the genitive case for its specification.
Dobili su set američkih knjiga na poklon.
They received a set of American books as a gift.
In this sentence, 'američkih' is the genitive plural form of the adjective 'američki', modifying 'ambasada' (embassies), which is also in the genitive plural to match. This usage denotes possession or association with 'American'. Example: the embassies belonging to or associated with Americans.