What made Allied leaders worry about the cost of invading the Japanese mainland?

The Japanese put great emphasis on the Bushido code, which was a guideline of morals a samurai would follow, although samurai didn't exist anymore by WW2 the Japanese military still taught it's soldiers this code, which kept most Japanese soldiers from surrendering, the problem was is that the Japanese would keep fighting no matter what even if it meant getting themselves killed, the U.S. was already taking heavy casualties invading Japanese islands, if they invaded the mainland of Japan the casualties would be even worse for both sides, to prevent thousands of American soldiers from dying nuking Japan was the only solution seen by the Allied leaders.

Explanation:

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declare war, appoint military officers, sign treaties, make alliances