The verb extirpate originally literally meant "to weed out by the roots." Now you'd use it when you want to get rid of something completely as if pulling it up by the root.
Use the verb extirpate when you mean to destroy completely or get rid of completely. You can try to extirpate all the bedbugs that came home with you from your vacation, but you will probably be afraid that some resisted the exterminator to munch on you later.