Rule 1: In order to sum or subtract two physical quantities the quantities must have the similar dimension. The final physical quantity has the similar dimensions. Physical quantities with the same dimension in different systems of units can be added or subtracted by multiplying one of the quantities by a unit conversion factor to get compatible units.
Rule 2: The product of two physical quantities results in a new physical quantity that has the sum of the exponents of the dimensions of the starting two quantities.
Rule 3: The division of one physical quantity by another results in a new physical quantity that has the dimension of the exponents of the first quantity minus the exponents of the second quantity.
Rule 4: Measuring the square root of a physical quantity results in a new physical quantity having a dimension with exponents half of the initial dimension.
Rule 5: The derivative of a physical quantity with respect to another physical quantity results in a new physical quantity with the exponents of the first dimension minus the exponents of the other dimension.
Example: Speed v has dimension L/T, length l has dimension L, time t has dimension T, then dv/dt has dimension L/T2 of acceleration.
Rule 6: The integral part of a physical quantity over the range of one another physical quantity results in a new physical quantity that has a dimension with the sum of the exponents of the two quantities.
Example: Speed v has dimension L/T, time t has dimension T, then integral v dt has dimension of length L.