Learn Turkish: Nasıl ödüyorsunuz, nakit mi kart mı?

Nasıl means "how". Just like in English, if a sentence already has a specific question word like "how", "what", or "who", you are asking for information, not a yes/no answer. Therefore, you don't use the yes/no question particle mı for that part of the sentence.

Why does the verb ödemek (to pay) become ödüyorsunuz instead of "ödeyorsunuz"?

The verb stem is öde-. When adding the present continuous suffix -yor, a stem ending in a vowel must narrow that final vowel to fit vowel harmony. The e changes to ü, giving us ödüyor before we add the formal/plural "you" ending -sunuz.

Why use mi and mı at the end instead of a word for "or"?

When offering a direct choice between two things, Turkish usually places a yes/no question particle after each option. Nakit mi kart mı? is the standard way to ask "Cash or card?" (literally "Is it cash? Is it card?"). This sounds much more natural for questions than using a literal word for "or" like veya.

Why is the ending -sunuz used here?

The suffix -sunuz is for the plural "you" or the formal "you". Service workers like cashiers or waiters will almost always use this polite form when speaking to a customer, even if they are only talking to one person.