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lick (something) up
To clean, eat, or drink a substance by licking with the tongue. Usually said of an animal. The dog came over and licked up the spilled pasta sauce. Don't lick that up off the table—you're not a cat, for crying out loud!
lick something up
to clean up all of some substance by licking with the tongue. (Usually said of an animal.) Don't worry about the spilled milk. The dog will lick it up. The dog licked up the milk.
To eat or drink something completely by licking: The kitten licked up the milk. The dog licked the spilled food up.