Bombs rapidly release energy, which heats up the air or material surrounding them, causing an explosion. This happens due to changes of state happening between atoms, releasing energy in the process. With a nuclear bomb, the change of state happens within atoms instead, allowing for more energy to get stored and releasing a lot more energy compared to the reaction inside a non-nuclear bomb.
The famous equation is E=mc[2]. Energy is the mass times the speed of light, squared. Now the speed of light is huge, so not a lot of mass is required to get huge amounts of energy. Adding hydrogen to hydrogen in the sun gives helium, but there is actually a tiny difference in mass
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