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.

'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 remove
'entfernen' and 'herausnehmen' can both mean that something is taken away from its place. 'herausnehmen' is more precise when something is inside a container, packaging, or a larger whole and you take it out. 'entfernen' is more general and can also mean that you get rid of something, even if it was not inside something before. Learners may confuse the words because both are often used with objects. A simple guideline is: if something is inside, 'herausnehmen' fits very well; if something simply needs to be gone, 'entfernen' fits better.

Vor dem Termin nimmt die Ärztin die Spritze aus der Verpackung heraus.
Before the appointment, the doctor takes the syringe out of the packaging.
Im Kurs entfernt ihr bitte alle falschen Wörter aus dem Text.
In the course, please remove all incorrect words from the text.