because it is the object of the sentence and so der becomes den

Yeah but remember the umlaut (the 2 dots) ontop of the a. That makes it totally different. The german word for cat is katze. They are also different genders, kaese being male and katze being female.

There's a definite article in the German ("den"), so you need "the" in the English translation.

Because it says, more specifically, DEN Käse. Therefore, THE cheese.

You use it when a masculine noun is the object of the sentence, as in the verb is doing something to it.

Because you do not learn the numbers lesson until later and they want you to focus on what the current lesson is.

it is.

Feminine, neuter, and plural nouns are the same in the accusative as in the nominative.

die Katze sieht die Maus (feminine)

das Mädchen sieht das Messer (neuter)

die Männer essen die Äpfel (plural)

because Kaese is actually a masculine noun.

Not sure what you are asking. Ä / ä is just a letter in the alphabet; it's not a word by itself.

First: it is "isst" (eats), not "ist" (is). ;) Second: the accusative (4th) case of "das" is "das", not "den".

So: der - den das - das die - die die - die (plural)

You combined the indefinite article [for non-specific singular nouns] 'a' with the plural 'women'. 'Eine Frau' is singular: 'a woman'. This is the reason your answer wasn't accepted, though Duo's preference for 'lady' is odd.

'Woman' is the primary meaning of 'Frau', but it can also be translated as 'female', 'Mrs./Ms./Miss' or 'lady'. However, 'Dame' is the more appropriate -- and far more polite -- word for 'lady'.

Technically yes, I think so. But I don't think you would say "a wife" in English, it would always be "my wife" / "your wife" / "his wife" etc.

Frau is still translated both as lady and woman. But you have to be careful about typing womAn and not womEn.

In this sentence, you DO need the definite article the. Den Käse = 'the cheese', not just cheese generally.

Roughly: "the woman" is a woman who is known to the listener -- perhaps because you had spoken about her already.

"a woman" is a woman who has not been spoken about before. She is new to the conversation. Her identity is perhaps not important.

The distinction between eine Frau and die Frau in German is similar to that between "a woman" and "the woman" in English.

There are accepted sentences that use "a woman" and others that use "one woman".

Both are reasonable translations of eine Frau.

it keeps telling me i am wrong but correct at the same time.

Is it telling you something about "another correct answer: [exactly what you wrote]"? If so, try to ignore that and just consider it a kind of "well done, you got it correct."

It has nothing to do with the subject, Mann or Frau. Rather, use of the indefinite article (in the English translation) depends on what they’re eating. “Apple” is a countable singular noun that starts with a vowel, and takes the article “an”. “Cheese” is an uncountable noun that starts with a consonant, and takes no indefinite article at all.

isst can mean "eats" or "is eating" -- German doesn't make that difference.

So you can translate Die Frau isst einen Apfel as "the woman eats an apple" (every day) or "the woman is eating an apple" (right now).

Yes, the German sentence -- in the absense of context, like here -- could translate to either of those English sentences.

eine Frau isst Käse

Because it's at the beginning of the sentence.

Actually it's "is having" if followed by food means "is eating" not vice versa