President Truman's decision to drop the first atomic bomb was primarily influenced by two key factors:
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To destroy Japan's power to make war: The use of the atomic bomb was aimed at bringing a swift end to World War II by compelling Japan to surrender unconditionally. Truman and his advisors believed that the bomb would significantly weaken Japan's military capabilities and force them to capitulate.
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To justify the expense of the Manhattan Project: The Manhattan Project was an enormous investment of resources and time, and there was a desire to demonstrate the efficacy and value of the atomic bomb. Utilizing the bomb would validate the project and the sacrifices made during its development.
It is important to note that the decision was also influenced by considerations of minimizing further American casualties that would result from a land invasion of Japan, as well as geopolitical factors related to post-war influence, particularly in relation to the Soviet Union. However, the two points listed above capture the immediate reasons for the bomb's use.