-아/어 내다 means "manage to" or "succeed in doing" - completing something despite difficulty.
Accomplishment - 끝까지 해냈어요 (managed to finish)
Overcoming difficulty - Success after struggle
Persistence - Emphasizes effort to completion
Implies difficulty overcome
Positive achievement - Different from -아/어 버리다 (which can be regrettable)
I managed to pass the difficult exam.
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