In a manner showing frustration, irritation, or annoyance; in an exasperated way.
From 'exasperated' (past participle of exasperate) + '-ly' (adverb-forming suffix); compound formation creating an adverbial form expressing frustrated emotion.
When someone throws up their hands and sighs 'exasperatedly,' you know they've gone past merely annoyed into a zone where nothing seems to be going right and they're done trying to hide it.
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