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.
Probably not the first day, honestly, but it wouldn't have taken long, especially once the USSR joined the ground forces. The US managed to invade Okinawa without killing enormous numbers of civilians - the one thing you can say about the US ground forces during WWII is that they did their best to minimize civilian casualties. High command didn't care about that, though, and just wanted to show off their new bomb.