I don't think this sentence is in any way special and demands special treatment in Duolingo. I don't know why the course creators didn't use a pronoun here.
In general, pronouns are often dropped in colloquial speech, but rarely dropped in written texts and in formal language. Maybe this sentence sounds colloquial so that's why the pronoun is dropped? After all, we don't often ask about cutting potatoes in a formal way... That's my guess.
Карто́шки can be either a accusative plural form, or a genitive singular form.
It’s not plural because «карто́шка» is an uncountable mass noun. It behaves like «rice» in English.
But if you wanted to use a genitive singular, you can. «Можете нарезать карто́шки?» sounds OK with genitive too. Accusative shifts the meaning towards 'all the potatoes', while Genitive shifts the meaning towards 'some potatoes'.
Yes, I was corrected in some other topic.
I personally don't use it in plural, but such option indeed exists.
You can't use «can I» because «мо́жете» is a 2nd person plural form, so it can only be used with «вы» (but plural form can be used to address one person for politeness). So, «можете» implies «вы» and you can't translate it "can I", "can I" is «могу я?».
As for accepting or not accepting pronouns: generally, pronouns are needed in written and formal speech, but can often be dropped in dialogues and in informal situations. Maybe this phrase is written without a pronoun because it's something you often hear and rarely see written, so that's why it's written with a dropped pronoun.
Accusative Singular. You can find all the declensions of картошка here. https://en.wiktionary.org/wiki/%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D1%88%D0%BA%D0%B0 This of course does mean either the English or Russin sentence is, as of right now, not a correct translation.
I would say «мо́жете поре́зать карто́фелину»? (But if precision is not required, you can use the «карто́шку» for one potato too. I'm not sure about the verb, for some reason «поре́зать» sounds slightly better to my ear when speaking about one potato, but it might be just me.)
Well, I try to write the standard forms in Duolingo, but in this exact case I just didn't know my stress is non-standard :)
Upd.: I've updated my message.
According to my conjugation table, both perfective and imperfective aspects of the infinitive are spelled the same, but the stressed syllable is different: нареза́ть (imperfective) / наре́зать (perfective).
While I realize the computer voice is far from accurate, at least for this sentence, I hear "her" saying the perfective aspect infinitive. Would that be correct?
Yes, that’s correct. She is pronouncing «наре́зать», and this sentence sounds natural with «наре́зать». «Мо́жете наре́зать» is a question about slicing a certain number of potatoes.
(«Мо́жете нареза́ть» could work if you’re asking to slice potatoes for some period of time, and it doesn’t matter if you slice all of them or not; or if you’re asking to be slicing potatoes repeatedly, not just one time.)
From what I understand, it can be singular, plural OR mass. So "potatoes" and "potato" should both be fine.
However, looking at the context of the sentence, notice that the subject is plural (because можете). How likely is it that someone is asking a group of people to slice one single potato? ;)