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Answer 1 · 2017-06-03T11:32:15

Dimension of the velocity of light, c #=[L][T]^-1#

Dimension of the Gravitational constant, G #=[M]^-1[L]^3[T]^-2#

Dimension of Planck's constant, h #=[M][L]^2[T]^-1#

Let the dimension of mass in new system be

#[M]=[c]^x [G]^y [ h]^z#

Hence

#[M]=([L][T]^-1)^x ([M]^-1[L]^3[T]^-2)^y] ([M][L]^2[T]^-1)^z#

#=>[M][L]^0[T]^0=[M]^(z-y) [L]^(x+3y+2z) [T]^(-x-2y-z)#

so comparing the power of both sides we have

#z-y=1......[1]#

#x+3y+2z=0......[2]#

#-x-2y-z=0.....[3]#

Adding [2] and [3]

#y+z=0.....[4]#

Adding [1] and [4]

#2z=1=>z=1/2#

so #1/2-y=1=>y=-1/2#

Inserting y and z in [2]

#x+3(-1/2)+2xx1/2=0#

#=>x-3/2+1=0#

#=>x=1/2#

Hence

#[M]=[c]^(1/2) [G]^(-1/2) [ h]^(1/2)#