
abarbeiten
to work through
To deal with a series of tasks or a large amount of work systematically and one after another.
Ich muss heute noch eine lange Liste von E-Mails abarbeiten.
I still have to work through a long list of emails today.
to work through / work off
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to work through
To deal with a series of tasks or a large amount of work systematically and one after another.
Ich muss heute noch eine lange Liste von E-Mails abarbeiten.
I still have to work through a long list of emails today.

to wear oneself out
To completely exhaust oneself through excessive physical or mental effort, often without achieving the desired result.
Er hat sich an dem Projekt völlig abgearbeitet und ist jetzt krank.
He completely wore himself out on the project and is now ill.

to work off
To settle a debt through work instead of paying with money.
Früher mussten die Knechte ihre Schulden beim Bauern abarbeiten.
In the past, farmhands had to work off their debts to the farmer.
'Abarbeiten' is a separable verb. The prefix 'ab-' goes to the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Sie arbeitet die ganze To-do-Liste ab.
She works through the whole to-do list.
In the meaning 'to deal with systematically,' 'abarbeiten' is often combined with things that have a structure. Typical objects are: a list, a plan, a stack of files, emails, or agenda items.
Zuerst müssen wir die dringendsten Kundenanfragen abarbeiten.
First we have to work through the most urgent customer inquiries.
The reflexive form 'sich abarbeiten' means to exert oneself to the point of exhaustion. The preposition 'an' with the dative is often used to name the object of the effort: 'sich an etwas abarbeiten' (to wear oneself out on something).
Sie hat sich an diesem schwierigen Problem völlig abgearbeitet.
She completely wore herself out on this difficult problem.
The meaning 'to pay a debt through work' is rarer today and often historical. You can recognize it from the object of the sentence, which is usually 'Schulden' (debts) or 'eine Schuld' (a debt).
Er musste seine Schulden auf dem Hof des Großbauern abarbeiten.
He had to work off his debts on the large farmer's estate.
to take care of / to finish
Both words can express completing several tasks. „abarbeiten“ emphasizes working through a set of tasks systematically and one after another; „erledigen“ simply says that the tasks are completed. „abarbeiten“ is more suitable for a list or a large amount of work, while „erledigen“ is more suitable for a single task.

Ich arbeite die Aufgaben auf meiner Liste nacheinander ab.
I’ll work through the tasks on my list one by one.
Ich erledige die wichtigste Aufgabe zuerst.
I’ll take care of the most important task first.