kauen
to chew
To break down food with the teeth before swallowing it.
Man sollte jeden Bissen gut kauen, um die Verdauung zu erleichtern.
You should chew every bite well to make digestion easier.

to chew
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to chew
To break down food with the teeth before swallowing it.
Man sollte jeden Bissen gut kauen, um die Verdauung zu erleichtern.
You should chew every bite well to make digestion easier.

The literal meaning (breaking down food) is often used without a preposition or with 'auf'. The figurative meaning (having a problem to deal with) always appears in the fixed expression 'an etwas (Dativ) zu kauen haben' (to have something to chew on/to struggle with something).
Er kaute nachdenklich auf seinem Stift.
He thoughtfully chewed on his pen.
When you want to say what someone is chewing on, you use the preposition 'auf' followed by the dative.
Das Kind kaut auf seinem Spielzeug.
The child is chewing on his toy.
Common things people chew are 'Kaugummi kauen' (to chew gum) or, out of nervousness, 'Fingernägel kauen' (to bite/chew one's fingernails). When eating, people often emphasize that you should chew 'gut' (well) or 'gründlich' (thoroughly).
Hör bitte auf, an deinen Fingernägeln zu kauen.
Please stop chewing your fingernails.
The idiom 'an etwas zu kauen haben' (to have something to chew on/to struggle with something) is a fixed structure. The 'etwas' is in the dative and describes the problem that occupies someone mentally or emotionally.
Der Sportler hatte an der Kritik der Presse schwer zu kauen.
The athlete struggled a lot with the criticism from the press.
to swallow
'schlucken' and 'kauen' are both connected to eating, but they describe different steps. 'kauen' means breaking food down with your teeth. 'schlucken' means that something goes from the mouth into the throat and then further into the body. The words can be confused because both things happen directly one after the other. First you chew a piece of bread, then you swallow it. With drinks, you normally only say 'schlucken', not 'kauen'.

Das Kind kaut langsam, weil das Brot sehr hart ist.
The child is chewing slowly because the bread is very hard.
Sie hat die Tablette mit einem Glas Wasser geschluckt.
She swallowed the tablet with a glass of water.