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Ausgehen vs. aussteigen

The German verbs ausgehen and aussteigen both start with aus- (out), but they describe very different actions. In essence, ausgehen relates to going out for leisure or supplies running out, while aussteigen means getting out of a vehicle or quitting something.

Ausgehen

A2
This verb most commonly means to go out for a social or recreational activity, like going to a restaurant, a bar, or the cinema. It also has several other important meanings, such as to run out of something, to assume or suppose, or for a situation to turn out in a certain way.
Wollen wir am Samstag zusammen ausgehen?
(Do you want to go out together on Saturday?)
Mir ist das Geld ausgegangen.
(I have run out of money.)
Ich gehe davon aus, dass der Plan funktioniert.
(I assume that the plan will work.)
Wie ist die Prüfung für dich ausgegangen?
(How did the exam turn out for you?)
Plötzlich ist das Licht im ganzen Haus ausgegangen.
(Suddenly, the light in the whole house went out.)
Die Initiative ging von den Studenten aus.
(The initiative originated from the students.)

Aussteigen

A1
This verb's primary meaning is to get out of or to get off a form of transportation, such as a car, train, bus, or airplane. Figuratively, it is also used to mean to quit, to drop out of, or to withdraw from a project, an agreement, a company, or even society.
Wir müssen an der nächsten Haltestelle aussteigen.
(We have to get off at the next stop.)
Er half seiner Großmutter, aus dem Auto auszusteigen.
(He helped his grandmother get out of the car.)
Nach vielen Meinungsverschiedenheiten ist sie aus dem Projekt ausgestiegen.
(After many disagreements, she quit the project.)
Das Land droht, aus dem internationalen Abkommen auszusteigen.
(The country is threatening to withdraw from the international agreement.)
Immer mehr Menschen wollen aus dem stressigen Stadtleben aussteigen.
(More and more people want to drop out of the stressful city life.)
Der Fahrer bat die Passagiere, schnell auszusteigen.
(The driver asked the passengers to get off quickly.)

Summary

In summary, the core difference is the context of out. You ausgehen (go out) from your home for a fun activity. You aussteigen (step out) from a vehicle or a commitment. Think of it this way: you ausgehen to a restaurant, and you aussteigen from the bus that takes you there.