Consider this second-order reaction:

(A+B C and rate = k[A]2).

What will happen to the reaction rate if the concentration of A is doubled?

If the rate doubles when the concentration of reactant is doubled, the reaction is the first-order. If the rate becomes 4-fold times (quadruples) when the concentration of the reactant is doubled, the reaction is the second-order and similarly, the third, the fourth, and the higher orders.

The rate will quadruple. 2^2 = 4