Question

supporters of using the atomic bomb against japan believed it would
save american lives.
end the war in europe.
improve relations with china.
improve relations with russia.

Explanation:

Supporters of the atomic bomb's use against Japan argued that a full-scale invasion of the Japanese mainland would cause massive American military casualties. Using the bomb was seen as a way to force a quick Japanese surrender, avoiding this high loss of life. The war in Europe had already ended before the bomb was used, and the other options were not primary justifications from supporters.

Answer:

save American lives.