They dropped two nuclear bombs on Japan to make the Soviet Union's invasion of Hokkaido unnecessary and as a performative act to horrify the world in to subjugation.
The whole 'saved more lives than it cost' is a bad argument unless the plan was to slaughter a hundred thousand civilians on landing. Note the Nazi invasion of France cost 60,000 lives in civilian resistance...this was at the beginning of the war.
Japan would have surrendered on the first day of invasion.
most milhis nerds would probably say that the rkka had no capability of invading japan, so ACKSHUALLY the nukes made us surrender. this ignores of course their amphibious operations in the baltic and the ijn being the equivalent of a battleship and about 200 other artifical reefs around the pacific