Learn Russian: Я должен убирать комнату и мыть посуду. - I must clean the room and wash the dishes.

Должен is a short-form adjective used as a predicate meaning 'must' or 'have to'. Because it is an adjective rather than a standard verb, it must agree with the gender of the subject. Here, the speaker (я) is male. If a woman were speaking, she would change the ending and say Я должна.

Are комнату and посуду in the accusative case?

Yes! They are the direct objects receiving the action of the verbs убирать (to clean) and мыть (to wash). Their dictionary forms are комната and посуда. Feminine nouns ending in -а change to -у in the accusative case, even when they follow an infinitive verb.

Why use the imperfective verbs убирать and мыть instead of perfective ones?

Using imperfective verbs here suggests that this is a regular, repeating chore. The speaker is talking about their general, ongoing duties. If this was a one-time task that just needed to be completed right now, they would likely use the perfective verbs убрать and помыть.

The English translation says 'dishes', but isn't посуду singular?

Exactly. In Russian, посуда is a collective noun. It functions just like the English words 'tableware' or 'crockery'. It is always grammatically singular, even though it refers to multiple cups, plates, and bowls.