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You pretty much write out the rate equations for two equations. You then divide the equations from each other. You want to pick two equations where the concentration of one reactant will be left over. The rate constant will cancel and you will have only one variable left over, which is the order of that reactant. You do that procedure until you have the order of all the reactants. You can then use the given initial rate with the calculated order of reactants to find your rate constant.
To add on, this explanation is usually called the experimental method. Dr. Lavelle went over this in the lecture on 2/28, using an example with NH4+ and NO2-. If you need extra help, I can explain that specific example step by step :)
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