Using the example reaction aa + bb → cc + dd write the potential first order and second order rate laws

Given the equation aA + bB → cC + dD, which is an irreversible reaction, the most common rate laws are the product of the concentration of the reactants, each raised to a power.

This is the the general form of such laws.

r = K [A]^m * [B]^n

The order of the reaction is the sum of the exponents: m + n.

For first order reaction m+ n = 1, yields to the next potential laws:

m = 1, n = 0 => r = k [A]
m = 0, n = 1 => r = k [B]
m=0.5, n = 0.5 => r = [A]^0.5 [B]^0.5

And other combinations where the sum of the exponents are 1.

For second order reactions, m + n = 2, these are some potential laws:

m = 1, n = 1 => r = k[A][B]
m=0, n = 2 => r = k[B]^2
m=2, n = 0 => r = k[A]^2

m=1.5, n = 0.5 => r = k[A]^1.5 [B]^0.5

As you see there are many potential laws all of them with the form

r = [A]^m [B]^n, being the order of the reaction m + n