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.