annehmen
to accept
To accept or receive something that is offered to you.
Ich kann dieses teure Geschenk nicht annehmen.
I can't accept this expensive gift.

to accept / assume
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to accept
To accept or receive something that is offered to you.
Ich kann dieses teure Geschenk nicht annehmen.
I can't accept this expensive gift.

'Annehmen' (to accept / assume) is a separable verb. The prefix 'an-' goes at the end in a main clause, stays attached to the verb in a subordinate clause, and in the infinitive with 'zu', 'zu' goes in between.
Er nimmt das Paket für seine Nachbarin an.
He accepts the package for his neighbor.
If you want to express an assumption, the structure 'Ich nehme an, dass ...' (I assume that ...) is very common. The subordinate clause with 'dass' then contains the assumption.
Ich nehme an, dass die Besprechung um 10 Uhr beginnt.
I assume that the meeting starts at 10 o'clock.
The expression 'sich einer Sache annehmen' (to take care of something) requires the genitive. This is a fixed structure that occurs mainly in written language.
Der Bürgermeister will sich des Problems der alten Schule annehmen.
The mayor wants to take care of the problem of the old school.
In the meaning 'akzeptieren' (to accept), 'annehmen' (to accept) is often combined with things that you are offered, for example: to accept an offer, an invitation, help, or an apology.
Er hat ihre Entschuldigung sofort angenommen.
He accepted her apology immediately.
This meaning of 'annehmen' (to adopt) is used almost exclusively in formal, political, or legal contexts. Typical signal words are 'Parlament' (parliament), 'Gesetz' (law), 'Antrag' (motion/application), or 'Vorschlag' (proposal).
Der Antrag wurde mit großer Mehrheit angenommen.
The motion was adopted by a large majority.
When something acquires a new quality, 'annehmen' (to take on) is often combined with words like 'Farbe' (color), 'Form' (form), or 'Gestalt' (shape).
Der Teig muss eine feste Form annehmen, bevor er gebacken wird.
The dough must take on a firm shape before it is baked.
to take
'nehmen' and 'annehmen' can both mean that you accept something you receive. 'nehmen' is simpler and more general; you can take/accept money, help, or an offer. 'annehmen' places more emphasis on the fact that something was offered and that you say yes to it. It often sounds a little more formal or precise, especially with invitations, offers, or apologies. The words can be confused because they work similarly in some sentences. When the act of accepting is important, 'annehmen' fits better.

Nach der Nachricht nimmt Samir das Angebot des Verkäufers an.
After the message, Samir accepts the seller's offer.
Die Ärztin nimmt für die Behandlung kein Bargeld.
The doctor does not accept cash for the treatment.