
ausüben
to exercise
To carry out an activity, profession, or function regularly or officially.
Sie übt den Beruf der Anwältin seit zehn Jahren aus.
She has exercised the profession of lawyer for ten years.
to exercise / practice
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to exercise
To carry out an activity, profession, or function regularly or officially.
Sie übt den Beruf der Anwältin seit zehn Jahren aus.
She has exercised the profession of lawyer for ten years.

to exert
To apply power, influence, pressure, or an effect to someone or something.
Der Trainer übt großen Einfluss auf die jungen Spieler aus.
The coach exerts great influence on the young players.

to practice
To practice a hobby, sport, or art form regularly.
In seiner Freizeit übt er den Bogensport aus.
In his free time, he practices archery.
'Ausüben' (to exercise/practice) is a separable verb. The prefix 'aus-' comes at the end in a main clause. In the infinitive with 'zu', 'zu' comes between the prefix and the verb.
Er übt seinen Beruf mit großer Leidenschaft aus.
He exercises his profession with great passion.
The verb 'ausüben' (to exercise/practice) belongs to more formal language. It is often used in official, professional, or written contexts.
Der Präsident übt sein Amt gemäß der Verfassung aus.
The president exercises his office in accordance with the constitution.
In this meaning, 'ausüben' (to exercise) is combined with professions or official activities. Common objects are 'einen Beruf' (a profession), 'ein Amt' (an office), 'eine Funktion' (a function), or 'eine Tätigkeit' (an activity).
Sie wollte schon immer eine leitende Funktion ausüben.
She had always wanted to exercise a leadership function.
When you exert influence or pressure on someone or something, the preposition 'auf' is used with the accusative. The object is often an abstract noun such as 'Druck' (pressure), 'Macht' (power), or 'Einfluss' (influence).
Die Eltern üben Druck auf ihre Kinder aus, gute Noten zu schreiben.
The parents put pressure on their children to get good grades.
to operate / pursue
Betreiben and ausüben can both express regularly engaging in a hobby or sport. Betreiben tends to emphasize actively and continuously pursuing the activity, while ausüben focuses more on actually practicing it. Both words are often possible with a hobby; ausüben is especially suitable when the activity itself is the focus.

Er übt seit Jahren Klettern in einer Halle in seiner Stadt aus.
He has been practicing climbing at an indoor gym in his city for years.
Er betreibt seit Jahren Klettern als anspruchsvolles Hobby.
He has been pursuing climbing as a demanding hobby for years.