
herausnehmen
to take out
To remove something from a container, packaging, or a larger whole.
Sie nahm vorsichtig den heißen Kuchen aus dem Ofen.
She carefully took the hot cake out of the oven.
to take out / take the liberty
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to take out
To remove something from a container, packaging, or a larger whole.
Sie nahm vorsichtig den heißen Kuchen aus dem Ofen.
She carefully took the hot cake out of the oven.

to take the liberty
To presume to do or say something that is perceived as intrusive or inappropriate; to have the nerve to do something.
Was nimmst du dir eigentlich heraus, so mit mir zu reden?
What makes you think you can take the liberty of talking to me like that?
'Herausnehmen' (to take out) is a separable verb. The prefix 'heraus-' comes at the end in a main clause. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Er nimmt das Buch aus der Tasche heraus.
He takes the book out of the bag.
The prefix 'heraus-' describes a movement from inside to outside, typically toward the speaker. It makes clear that something is being removed from a surrounding space.
Kannst du bitte die Getränke aus dem Keller herausnehmen und nach oben bringen?
Could you please take the drinks out of the cellar and bring them upstairs?
You can easily distinguish the two main meanings. The literal meaning ('entfernen' — to remove) is not reflexive. The figurative, negative meaning ('sich anmaßen' — to presume) is always reflexive: 'sich etwas herausnehmen' (to take the liberty of doing something).
Sie nimmt die Wäsche aus der Maschine heraus.
She takes the laundry out of the machine.
In the meaning 'sich anmaßen' (to presume), the verb is always reflexive, and the pronoun is in the dative: 'sich (Dativ) etwas herausnehmen'. It expresses strong indignation about a person's behavior.
Ich kann nicht glauben, dass sie sich das herausgenommen hat.
I can't believe that she took the liberty of doing that.
to infer / take
Entnehmen and herausnehmen can both express removing an object from a container. Entnehmen usually sounds more formal and can emphasize a deliberate removal; herausnehmen is more everyday and makes the outward movement clearer.

Nach der Untersuchung nimmt die Ärztin die übrigen Proben aus dem Behälter heraus.
After the examination, the doctor takes the remaining samples out of the container.
Für die Untersuchung entnimmt die Ärztin dem Behälter gezielt eine Probe.
For the examination, the doctor deliberately takes a sample from the container.