If the velocity of light `(c )` , gravitational constant `(G)` , and Planck's constant `(h)` are chosen as fundamental units , then find the dimensions of mass in new system.

Explanation

Let `m prop c^(x) G^(y) h^(z) or m = K c^(x) G^(y) h ^(z)`
By substituting the dimension of each quantity in both the sides,
` [M^(1) L^(0) T^(0) ] = K [ LT^(-1)]^(x) [ M^(-1) L^(3) T^(2)]^(y) [ ML^(2)T^(-1)]^(z)`
`= [ M^(-y + z) L^( x + 3 y + 2z ) T^(-x - 2 y - z)]`
By equating the power of `M , L , and T` in both the sides : `-y + z = 1 , x + 3y + 2z = 0 , -x - 2y -z = 0` .
By solving above three equations , `x = 1//2 , y = -1//2 , and z = 1//2`.
:. ` m prop c ^(1//2) G^(-1//2) h^(1//2)`