sich ausdehnen
to expand
To increase in volume, area, or length; to become larger. Usually refers to physical processes.
Bei Erwärmung dehnt sich das Metall aus.
When heated, the metal expands.

to expand / extend
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to expand
To increase in volume, area, or length; to become larger. Usually refers to physical processes.
Bei Erwärmung dehnt sich das Metall aus.
When heated, the metal expands.

'Ausdehnen' (to expand/extend) is a separable verb. The prefix 'aus-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Das Gas dehnt sich bei Erwärmung schnell aus.
The gas expands quickly when heated.
This verb has a reflexive and a transitive use. 'Sich ausdehnen' means that something becomes larger by itself. 'Etwas ausdehnen' means that someone actively enlarges or extends something.
Die Wüste dehnt sich jedes Jahr weiter aus.
The desert expands further every year.
When something is actively extended, the objects are often abstract. Typical combinations are 'die Suche ausdehnen' (to extend the search), 'die Zusammenarbeit ausdehnen' (to expand the cooperation), or 'eine Diskussion ausdehnen' (to extend a discussion).
Die Polizei musste die Suche auf das Nachbardorf ausdehnen.
The police had to extend the search to the neighboring village.
When something spreads spatially, the preposition 'auf' or 'über' with the accusative is often used. This describes the target or area of the spread.
Der Stau dehnte sich über die gesamte Autobahn aus.
The traffic jam extended across the entire highway.
to expand
'erweitern' and 'ausdehnen' can both mean that something becomes larger. 'erweitern' is often used when you deliberately make something larger or add new parts, for example a range of services, a house, or knowledge. 'ausdehnen' works especially well when something becomes larger spatially, temporally, or physically, for example an area, an appointment, or a material when heated. The words can be confused because both express an increase in size. A practical guide: 'erweitern' often sounds like adding and planning; 'ausdehnen' sounds more like spreading, length, area, or duration.

Bei großer Hitze dehnt sich Metall aus.
In extreme heat, metal expands.
Der Verein erweitert sein Programm um Kurse für Kinder.
The club is expanding its program to include courses for children.