If you do something expeditiously, you do it with speed and great efficiency. You might work expeditiously to finish your chores so you can go out with your friends.
The adverb expeditiously can describe something you do quickly and efficiently, but the word has a formal sound to it that makes whatever you’re doing sound fancy. You might do the dishes quickly just to get it over with — but tell your mom you did them expeditiously and she’ll be more impressed. You’ll often hear the word used in formal contexts. A police spokesperson might talk about an investigation that will be handled expeditiously.