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I'm not sure if I'm fully correct, but a Joule derives from a Newton (kg.m/s^2). Joules measures amount of work done on an object (mass) going through a distance of one meter. Which is why a Joule is a Newton times mass, so kg.m^2.s^-2.
If you think about the equation for kinetic energy you see that Ek=(1/2)mv^2. (1/2) is a constant and has no units. Mass has units in Kg and velocity is measured in m/s. The equation asks you to square velocity and if you think about your exponent laws (m/s)^2 comes out to be m^2/s^2. This equation does not show divisibility, it alters the division of seconds and makes it a product by raising seconds to the negative power, and that is how you end up with J=Kgm^2s^-2. Hope that helped :)
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