Reasons the US bombed nuked Japan (in no particular order of importance)
1. To save US lives. The Japanese barely surrendered, their Bushido code of honour required them to fight to the death so if it came to invading Japan it would've meant huge casualties on both sides (going by previous engagements with the Japanese it would be a 1 to 5 ratio in favour of the US...and the Japanese were preparing to amass millions of troops for defense of their homeland). Their is only so much conventional air power can do though it did devastate Japan far more then the nukes did.
2. To show the world (and in particular the USSR) the US was the new world superpower and the only country in possession of the ultimate weapon.
3. To evaluate atomic weapons use in combat (or against civilians in this case).
4. Revenge.
5. Japan refused to surrender unconditionally.
I leave you with two quotes-
General Douglas MacArthur, Commander of US Army forces in the Pacific, stated on numerous occasions before his death that the atomic bomb was completely unnecessary from a military point of view: "My staff was unanimous in believing that Japan was on the point of collapse and surrender."
General Curtis LeMay, who had pioneered precision bombing of Germany and Japan (and who later headed the Strategic Air Command and served as Air Force chief of staff), put it most succinctly: "The atomic bomb had nothing to do with the end of the war."
