I just learned where the slang, "I don't give a rat's ass" originated from. History is interesting. The slang means, "not to care about something or someone at all".
During World War II, British weapons engineers tried to sabotage strategic enemy locations by filling dead rats with small amounts of explosives, and placing them near boiler rooms and in coal piles, where maintenance technicians would hopefully see the dead rat, and dispose of it by throwing it into the furnace, detonating the explosives and destroying enemy infrastructure.
The idea didn't pan out because German intelligence got wind of it, and trained everyone to check dead rats they may find as a safety measure, which included inspecting the rat's behind, something that suddenly became of critical importance.
Thus originated the phrase, "I couldn't give a rat's ass," when something is not important to you.