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They're all basically synonyms. They mean, essentially, "to free from blame".
However, they're used in slightly different contexts. "Acquit" is used in court, specifically to refer to the process by which a judge or jury removes official charges of a crime against a defendant. "Absolve" is used more in a religious sense, where a priest would "absolve someone of their sin" or something along those lines, or far less commonly in a general civil sense to remove a person of an obligation (e.g. "I absolve you from your promise to cut my grass"). And "exculpate" simply isn't used anywhere that I'm aware of, except maybe in court briefings.
