Derive the relation for conversion of one unit system to another using the dimensions.

Hint: In order to answer this question, to know the derivation of a given relation between two systems, we will take two systems of the physical quantities and then compare them by making their equations by their formula with their different units.

Complete step by step answer:
Conversion of physical quantity units from one system to another: When the value of a physical quantity in one system is known, the process of dimensional analysis may be used to determine its value in another system.
As discussed, the measurement of a physical quantity is given by:
$Q = nu$
If the unit of a physical quantity in a system is ${u_1}$ , and the numerical value is ${n_1}$ , then:
$Q = {n_1}{u_1}$ ……...…(i)
Similarly, in the other system if the unit is ${u_2}$ and magnitude is ${n_2}$ then:
$Q = {n_2}{u_2}$ ……...…..(ii)
From eq(i) and eq(ii):
${n_1}{u_1} = {n_2}{u_2}$ ……...……(iii)
If \[a,b,c\] are the dimensions of a physical quantity in mass, length and time in the two systems, then
${n_2}={n_1}{[\dfrac{{{M_1}}}{{{M_2}}}]^a}{[\dfrac{{{L_1}}}{{{L_2}}}]^b}{[\dfrac{{{T_1}}}{{{T_2}}}]^c}$
When the value of a physical quantity in the first system is determined, the equation can be used to find its value in the second or new system.

Note: A system of units is a set of similar units that are used in calculations. It includes both fundamental and derived units. Some units are used in many systems of measurement. The following are the various systems of units used to quantify physical quantities: The C.G.S (\[\text{Centimetre, Gram, Second}\]) unit system is a French system.