Learn German: Die Frau antwortet keinem Mann. - The woman is not answering a man.

The verb antworten (to answer) always requires the person you are answering to be in the dative case. Because Mann is masculine, the negative article kein gets the dative masculine ending -em, making it keinem.

Why does antworten take the dative case instead of the accusative?

While in English you "answer someone" directly, German treats this verb more like "giving an answer TO someone." The dative case often represents this "to" or "for" relationship, which is a helpful trick to remember why antworten uses it.

The translation says "not answering a man", so why use keinem instead of nicht?

A more literal translation would be "The woman answers no man." When you want to say "not a" in German, you almost always combine them into the negative article kein (or in this case, the dative keinem). It sounds much more natural than using nicht and an article separately.

How would this sentence change if she was not answering a woman?

Since woman (Frau) is a feminine noun, it takes the dative feminine ending -er. The sentence would become Die Frau antwortet keiner Frau.