If the velocity of light 'C', the gravitational constant 'G' and Planck's constant 'h' are chosen as the fundamental units,...

What an interesting question! In the current system of measurements, the fundamental units are the units of length [L] = meters, time [t] = seconds, and mass [M] = kilograms. (There is also the unit of electric charge, Coulomb, but this is not relevant to this question.) All other quantities are measured in the units that are composed from the fundamental units.

For example, velocity is measured in meters per second, m/s, force is measured...

What an interesting question! In the current system of measurements, the fundamental units are the units of length [L] = meters, time [t] = seconds, and mass [M] = kilograms. (There is also the unit of electric charge, Coulomb, but this is not relevant to this question.) All other quantities are measured in the units that are composed from the fundamental units.

For example, velocity is measured in meters per second, m/s, force is measured in Newtons: `1 N = (kg*m)/s^2` , and work is measured in Joules: `1J= (kg*m^2)/s^2` .

Now, we are going to assume that the units in which the speed of light c, gravitational constant G and Plank's constant h are measured are the fundamental units, and compose the units of measurement of mass, kilograms, from them.

The speed of light is `c = 3*10^8 m/s` ; [c] = m/s.

The gravitational constant is `G = 6.67*10^(-11) m^3/(kg*s^2)` ; `[G] = m^3/(kg*s^2)`

The Planck's constant is `h = 6.63*10^(-34) (m^2*kg)/s` ; `[h] = (m^2*kg)/s`

Note that the kilograms, the unit we need to express through [c], [G], and [h], is found in the denominator of [G] and the numerator of [h]. So the only way to produce kilograms while involving both is to divide one by another:

`([h])/([G]) = (m^2*kg)/s / (m^3/(kg*s^2)) = (m^2*kg)/s * (kg*s^2)/m^3`

Simplifying this results in `([h])/([G]) = (kg^2*s)/m` .

Notice that s/m is the reciprocal of the units of velocity: [c] = m/s, so

`([h])/([G]) = (kg^2)/([c])`

Now we can express the kilograms in terms of the new "fundamental" units:

`kg^2 = ([h]*[c])/([G])` and

`kg = sqrt(([h]*[c])/([G]))`

So the dimensions of mass will be

`([h]^(1/2) * [c]^(1/2))/([G]^(1/2))` .