If the velocity of light c, the constant of gravitation G . . .

Question : If the velocity of light c, the constant of gravitation G and plank's constant h be chosen as fundamental units, find the dimensions of mass, length and time in terms of c, G and h. 

Doubt by Arham

Solution : 

The dimensional formula of 
Speed of light (c) = [M⁰LT^-1]

Gravitational Constant (G) = [M^-1L³T^-2]

Planck Constant (h) = [ML²T^-1]

Let 

m ∝ c^aG^bh^c

m = kc^aG^bh^c — (1)

[ML⁰T⁰] = [LT^-1]^a [M^-1L³T^-2]^b [ML²T^-1]^c
[ML⁰T⁰] = [L^aT^-a] [M^-bL^3bT^-2b] [M^cL^2cT^-c]
[ML⁰T⁰] = [M^-b+c L^a+3b+2c T^-a-2b-c]

Comparing the powers of M, L and T both sides. 

-b+c=1 — (2)

a+3b+2c=0 — (3)

-a-2b-c=0 — (4)

Adding equation (2), (3) and (4)
2c=1

c=1/2
putting in equation (2)
-b+1/2=1
-b=1-1/2
-b=1/2
b=-1/2

putting the values of b and c in equation (4)

-a-2(-1/2)-1/2=0
-a+1-1/2=0
-a+1/2=0
-a=-1/2
a=1/2

putting back the values of a, b and c in equation (1)

m = kc^1/2G^-1/2h^1/2

Hence, the dimensional formula of mass in terms of c, G and h is [c^1/2G^-1/2h^1/2].

Similarly, you could find the find the dimensional formula of length and time. 
L=[ c^-3/2 G^1/2 h^1/2]
T=[c^-5/2 G^1/2 h^1/2]