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The rate law for a zero order reaction is rate=k. Can also be written as rate=k[A]^0 to more plainly see that it is zero. From this you can tell that the concentration of the reactant will have no affect on the rate of the reaction, as it is solely dependent on the rate constant k.
Also, when looking at a zero order reaction in terms of mechanisms, they are not present in the reactant portion and only the intermediates are in that portion (if you wanted to look at it from a mechanism standpoint).
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