The 3 is pronounced like the final sound of the word "duh!" when you want to emphasize that the other is an idiot. Maybe you don't recognize it as its own sound and think of it like a part of the vowel. But its the sound that makes your throat vibrate a lot, since that's the part of the body that makes that sound.
The nominative case is a fancy way to call the form of a noun that is used for the subject of sentences in most languages. English does not have such a special form since it uses word order to indicate who is the subject, but other languages do, like German and Latin. The only exception in English are pronouns. "I, you, he, she, it, we, they" are the nominative forms of the English pronouns, "me, you, him, her, it, us, them" are the accusative forms (used to indicate objects) and "my, your, his, her, its, our, their" are the genitive forms (used to indicate possessors) (English also has a genitive for nouns: 's like in "Peter's"). This makes a helpful trick: if you want to know the case of some word, substitute it with a pronoun and look what list it belongs to in English. So, let's imagine you want to know the case of "Mary" and "apple" in "Mary eats an apple". We only can substitute those words with "She eats it". "She" is nominative, so Mary also is nominative. "It" is ambiguous, but we can use another trick: make it plural and the case is preserved. So "She eats them". "Them" is clearly accusative, so "an apple" is also accusative. So there you have it, not only nominative, but all three arabic cases.
Nominative case normally means the word is the subject of a sentence, the thing doing the action. It has "u" as the final vowel.
In arabic you say "dajaajun 3aadiyyun" if "regular chicken" is the subject and it's doing something, but "dijaajan 3aadiyyan" if it's the object, like if you're eating it.
So "ana akal dajaajan 3aadiyyan," is "I ate regular chicken," but "Regular chicken is good," is "dajaajun 3aadiyyun jayyid".
To say "THE regular chicken" you take off the "n" sound at the end, which isn't necessarily written, and when it IS written it's by addind an extra vowel mark, rather than the letter ن .
AND the final "n" and the vowel that comes before it aren't normally pronounced at all in dialectical speech, so the computer mispronounces things a lot. To get it right, you must make sure that nouns and the adjectives describing them agree and have the same ending.
So dajaaj 3aadiyy is how you would normally say it, but in formal speech it's dajaajun/an/in 3aadiyyun/an/in.
As I pointed in my response to AnthPalm:
"The 3 is pronounced like the final sound of the word "duh!" when you want to emphasize that the other is an idiot. Maybe you don't recognize it as its own sound and think of it like a part of the vowel. But its the sound that makes your throat vibrate a lot, since that's the part of the body that makes that sound."
Maybe this video and diagram could help?