Learn Icelandic: Við erum í ræktinni með honum.

In this sentence, we are doing something together with him. When the preposition með means "with" in the sense of accompaniment (being together), the word that follows it must be in the dative case. honum is the dative form of hann (he/him).

Does the preposition í also force the word after it into the dative case?

Yes! When í (in) describes a fixed location where someone or something already is, it takes the dative case. Because we are already located inside the gym, the word for "the gym" must change to its dative form.

How does the word ræktinni get its ending?

The base word for "gym" is the feminine noun rækt. To say "the gym", you add the feminine definite article to the end, making it ræktin. Because it needs to be in the dative case after the preposition í, that article changes to -inni, giving us ræktinni.

I've seen the word líkamsrækt used for gym. Is ræktinni a shortened version of that?

Exactly! líkamsrækt is the full, formal word for a gym or fitness center. However, in everyday spoken Icelandic, people almost always shorten it to just ræktin (the gym).