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Since there is no air and space, rockets need to take oxygen with them into space.  Inside the rocket's engine, fuel and oxidizers are ignited in the combustion chamber, creating hot, expanding gases.  That gas has pressure, and it pushes harder against the top of the rocket than the bottom, making it go up or forward.  Since the fuel is burning, exhaust is released out the bottom.  You can check out an example of how the Saturn V rocket engine works in this Explainer video.