
betrieben
powered
Driven or kept running by a particular energy source or mechanism.
Die alte Mühle wird heute elektrisch betrieben.
The old mill is powered electrically today.
powered / operated
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powered
Driven or kept running by a particular energy source or mechanism.
Die alte Mühle wird heute elektrisch betrieben.
The old mill is powered electrically today.

operated
Managed, run, or maintained by a person, group, or organization.
Das kleine, von einer Familie betriebene Hotel ist sehr gemütlich.
The small, family-operated hotel is very cozy.
'Betrieben' (powered/operated) is the past participle of the verb 'betreiben' (to operate/run). It is very often used like an adjective to describe a noun more precisely.
Die mit Wasserkraft betriebene Anlage ist sehr umweltfreundlich.
The facility powered by hydropower is very environmentally friendly.
The preposition reveals the meaning: 'betrieben mit' (powered by/with) describes the energy source, while 'betrieben von' (operated by) names the person or organization that runs something.
Die Heizung wird mit Gas betrieben.
The heating system is powered by gas.
In the meaning 'angetrieben' (powered), the word is often combined with energy sources. Common forms are: 'elektrisch betrieben' (electrically powered), 'solar betrieben' (solar-powered), 'mit Gas betrieben' (powered by gas), or 'mit Atomkraft betrieben' (powered by nuclear energy).
Immer mehr Fahrzeuge werden rein elektrisch betrieben.
More and more vehicles are powered purely electrically.
When it is about running a business or organization, typical combinations are: 'staatlich betrieben' (state-operated), 'privat betrieben' (privately operated), or 'von einer Familie betrieben' (family-operated).
Der Nationalpark wird staatlich betrieben und geschützt.
The national park is state-operated and protected.
to converse / entertain / maintain
Both words can refer to a facility that an organization looks after on an ongoing basis. Unterhalten emphasizes maintenance or funding, while betrieben refers to day-to-day management and organization. When the focus is on care and upkeep, unterhalten is appropriate; when the focus is on ongoing operation, betrieben is more appropriate.

Das Jugendzentrum wird von der Stadt betrieben und täglich organisiert.
The youth center is operated by the city and organized on a daily basis.
Die Stadt unterhält das Jugendzentrum und bezahlt notwendige Reparaturen.
The city maintains the youth center and pays for necessary repairs.