die Mühe
effort
The effort or expenditure of physical or mental energy required to complete a difficult task.
Die ganze Mühe hat sich am Ende gelohnt.
All the effort was worth it in the end.

effort / trouble
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effort
The effort or expenditure of physical or mental energy required to complete a difficult task.
Die ganze Mühe hat sich am Ende gelohnt.
All the effort was worth it in the end.

'Die Mühe' (effort/trouble) belongs to a word family with the adjectives 'mühsam' (laborious/arduous) and 'mühevoll' (painstaking/laborious). All of these words describe an activity that requires a lot of effort and energy.
Der Aufstieg auf den Berg war sehr mühsam.
The climb up the mountain was very arduous.
'Mühe' (effort/trouble) is often used in fixed expressions. Very common ones are 'sich Mühe geben' (to make an effort), 'sich die Mühe lohnen' (the effort was worth it), and to manage something 'ohne Mühe' (without effort/easily).
Wenn du dir in der Schule Mühe gibst, wirst du gute Noten bekommen.
If you make an effort at school, you will get good grades.
The expression 'sich die Mühe machen' (to take the trouble) is reflexive and is often expanded with an infinitive construction with 'zu'. The person making the effort does it for themselves, so 'sich' is in the dative.
Sie hat sich die Mühe gemacht, einen Kuchen für mich zu backen.
She took the trouble to bake a cake for me.
The expression 'Mühe haben' (to have difficulty) is often used with an infinitive construction with 'zu' or with the preposition 'mit' + dative. It expresses that an action is difficult.
Ältere Menschen haben oft Mühe, die moderne Technik zu verstehen.
Older people often have difficulty understanding modern technology.
work
'die Arbeit' and 'die Mühe' can both express the work, effort, or exertion involved in something. 'Arbeit' often sounds more factual: a task takes time and energy. 'Mühe' emphasizes more strongly that something is strenuous, difficult, or done with special care. For example, you say that something 'viel Arbeit macht' (takes a lot of work) when there is a lot to do. You say that someone 'sich Mühe gibt' (makes an effort) or 'sich Mühe macht' (takes the trouble) when they make a special effort. 'Mühe' also fits when someone has difficulty with something.

Sie gibt sich große Mühe, die Übungen richtig zu machen.
She makes a great effort to do the exercises correctly.
Die Vorbereitung für das Fest war viel Arbeit für die ganze Familie.
Preparing for the celebration was a lot of work for the whole family.