
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.

to extend
To lengthen something in time or expand it spatially or in terms of content.
Wir mussten die Diskussion auf den nächsten Tag ausdehnen, weil die Zeit nicht reichte.
We had to extend the discussion to the next day because there wasn't enough time.

to spread
To extend over a larger area or range; to expand.
Das Feuer drohte, sich auf den gesamten Wald auszudehnen.
The fire threatened to spread to the entire forest.
'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 usually emphasizes adding new components to a program or body of content. Ausdehnen places more emphasis on extending the scope, duration, or spatial area. Use erweitern for additional elements and ausdehnen when the main point is that the time span or reach becomes greater.

Das Museum dehnt sein Ferienprogramm auf drei Wochen aus.
The museum is extending its holiday program to three weeks.
Das Museum erweitert sein Ferienprogramm um einen zusätzlichen Workshop.
The museum is expanding its holiday program to include an additional workshop.