"have already been eaten" is more idiomatic in english here. or "were already eaten" i guess
So the past participle in Swedish is different from the third form of the verb, isn't it? Are there any rules of how it is formed?
Yes, it depends on whether the verb is weak or strong. The strong verbs are the ones with a vowel alternation, like springa sprang sprungit and the weak ones are the regular ones like prata pratade pratat. The participle of strong verbs are formed with -en based on the 3rd form, so the past participle of springa would be sprungen. You can compare this with e.g. ’taken’ in English, tagen in Swedish.
The weak ones depend on their verb class, but they take -ad if they’re class 1, so prata pratade pratat becomes pratad, whereas stänga stängde stängt (class 2) takes -d (stängd) if it ends on a voiced consonant but -t if it ends on a voiceless so köpa köpte köpt becomes köpt. This is similar to the past tense form.
Since past participles act as adjectives they also have to change after gender and number. Strong verbs like sprungen becomes sprunget in the neuter and sprungna in the plural or the definite form. Weak verbs like pratad becomes pratat in the neuter and pratade in the plural or definite. Note that it’s not the normal -a plural ending here.
If the verb is a particle verb, like springa upp, the particle moves to the front in the participles: uppsprungen.
This verb is äta upp, so the participle is äten, then it’s in the plural so ätna and the particle moves to the front so uppätna.
So for a word like "köpa", common and neuter gender are both "köpt" but plural is "köpta"?
Tack! This really explains a lot, and it's actually much easier than I thought it would be (no extra -a- for -er verbs as usual, and -en for strong verbs being a simple rule and not a thing to remember)
page 483 of this pdf has a good table that may be helpful: https://fsi-languages.yojik.eu/languages/FSI/Swedish/Basic/FSI%20-%20Swedish%20Basic%20Course%20-%20Student%20Text.pdf
cookie/biscuit = kaka (birthday) cake = (födelsedags)tårta
But for the block of desert, like cheese cake; it's kaka in Swedish. Don't get fooled by that.
Depends on what you mean. uppäten is a passive participle that works as an adjective, so you could say that this already is passive.
With äta upp, you can only use the verb separately. So when you're not using the participle, you can say kakorna har ätits upp. We don't use passive that much for concrete things though so I'd recommend learners not to use that kind of expression unless you're speaking about abstract things.