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Basically you take the rate law (rate=k[a] for first order) and the units of rate are always mol.L-1.s-1 and then the units of concentration are always mol.L-1 and then you plug those in to find units of k, which would be s-1.
Rate = k[A]^n, where n is the exponent that represents order of the reaction. Concentration is always going to have units of M/L, which is the equivalent to mol.L-1.s-1. Be sure to apply the exponent to the units of the concentration as appropriate. You want to isolate k to be alone on one side of the equal sign to find its units. Since the units of rate are always mol.L-1.s-1, you can divide the units of the rate by the units of [A]^n.
I know this question has been answered many times, but the simplest response is just to do dimensional analysis and multiply everything out. The units of k will be whatever makes the equation work out, so they differ according to the order of the reaction.
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