operieren
to operate
To perform a surgical procedure on a patient in order to cure an illness or treat an injury.
Der Arzt wird den Patienten morgen am Herzen operieren.
The doctor will operate on the patient's heart tomorrow.

to operate
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to operate
To perform a surgical procedure on a patient in order to cure an illness or treat an injury.
Der Arzt wird den Patienten morgen am Herzen operieren.
The doctor will operate on the patient's heart tomorrow.

The verb 'operieren' (to operate) in German describes actions on people (medical) or strategic actions (military, business). It is not used for the technical control of devices or machines.
Die Sondereinheit operiert nur nachts.
The special unit operates only at night.
In the medical meaning, the structure 'jemanden an etwas (Dativ) operieren' (to operate on someone on something) is often used to name the affected body part. The subject is usually medical personnel.
Der Chirurg musste ihn am Knie operieren.
The surgeon had to operate on his knee.
In the strategic meaning, 'operieren' (to operate) is often combined with adverbs that describe the manner of the action. In this case, it does not have a direct object.
Die Firma operiert international.
The company operates internationally.
In the scientific or mathematical meaning, 'operieren' (to operate) is almost always used with the preposition 'mit' (with) and the dative to name the concepts or symbols used.
Ein guter Programmierer operiert mit logischen Strukturen.
A good programmer operates with logical structures.
to work
'arbeiten' and 'operieren' can overlap when someone works with certain tools, numbers, or ideas, or acts within a specific field. 'arbeiten' is the normal and broad word. You can work on a project or work with data. 'operieren' sounds more specific and often more technical. It fits, for example, in mathematics, logic, business, or strategic contexts. Learners may confuse the words because 'mit Zahlen operieren' (to operate with numbers) is similar to 'mit Zahlen arbeiten' (to work with numbers). In everyday language, however, 'arbeiten' is usually more natural. Choose 'operieren' when the context is technical, strategic, or very precise.

In dieser Aufgabe operieren die Schüler mit einfachen Zahlen und Symbolen.
In this exercise, the students operate with simple numbers and symbols.
Wir arbeiten gerade an einem neuen Plan für das Team.
We are currently working on a new plan for the team.