If velocity of light c, Planck's constant h and gravitational constant G are taken as fundamental quantities, then express mass, length and time in terms of dimensions of these quantities.

Explanation

We know that, dimensions of `(h)=[M^(1)L^(2)T^(-1)]` (From E=hf)
dimensions of `(c) =[L^(1)T^(-1)](c)` (c is velocity)
Dimensions of gravitational constant
(G) `=[M^(-1)L^(3)T^(-2)] " " ("From" F=(Gm_(1)m_(2))/(r^(2)))`
(i) Let `m prop c^(a)h^(b)G^(c)`
`rArr m=kc^(a)h^(b)G^(c)`
where, k is a dimensionless constant of proportionality.
Substituting dimensions of each term in Eq. (i), we get
`[ML^(0)T^(0)]=[LT^(-1)]^(x)xx[M^(2)L^(-1)]^(y)[M^(-1)K^(-3)T^(-2)]^(z)`
`=[M^(b-c)L^(a+2b+3c)T^(-a-b-2c)]`
Comparing powers of same terms on both sides, we get
`b-c=1 " "...(ii)`
`a+2b+3c=0 " "...(iii)`
`-a-b-2c=0" "...(iv)`
Adding Eqs. (i), (iii) and (iv), we get
`2b=1 rArr b=(1)/(2)`
Substituting value of b in eq. (i), we get
`c=-(1)/(2)`
From eq. (iv)
`a=-b-2c`
Substituting values of b and we get
`a=-(1)/(2)-2(-(1)/(2))=(1)/(2)`
Putting values of a, b and c in equ. (i), we get
`m=kc^((1)/(2))h^((1)/(2))G^(-(1)/(2))`
`:.m=ksqrt((ch)/(G))`
(ii) Let `L prop c^(a)h^(b)G^(c)`
`:.L=kc^(a)h^(b)G^(c) " "..(iv)`
Where, k is a dimensionless constant.
Substituting dimensions of each term in equ. (v), we get
`[M^(0)LT^(0)]=[LT^(-1)]^(a)xx[ML^(2)T^(-1)]^(b)xx[M^(-1)L^(3)T^(-2)]^(c)`
`=[M^(b-c)L^(a+2b+3c)T^(-a-b-2c)]`
On comparing powers of same terms, we get
`b-c=0 " "...(vi)`
`a+2b+3c= " "...(vii)`
`-a-b-2c=0 " "...(viii)`
Adding equ. (vi), (vii) and (viii), we get
`2b=1`
`:.b=(1)/(2)`
Substituting value of b in equ. (vi), we get
`c=(1)/(2)`
From equ. (viii),
`a=-b-2c`
Substituting values of b and c, we get
`a=-(1)/(2)-2((1)/(2))=-(3)/(2)`
Putting values of a, b and c in equ. (v), we get
`L=kc^((3)/(2)h^((1)/(2))G^((1)/(2))`
`L=ksqrt((hG)/(c^(3)))`
(iii) Let `T prop c^(a)h^(b)G^(c)`
`:.T=kc^(a)h^(b)G^(c)`
where, k is a dimensionless constant, Substituting dimensions of each term in equ. (ix), we get
`[M^(0)K^(0)T]=[LT^(-1)]^(a)xx[ML^(2)T^(-1)]^(b)xx[M^(-1)L^(3)T^(-2)]^(c)`
`=[M^(b-c)L^(a+2b+3c)T^(-a-b-2c)]`
On comparing powers of same terms, we get
`b-c=0 " "...(x)`
`a+2b+3c=0 " "...(xi)`
`-a-b-2c=1 " "...(xii)`
Adding equ. (x). (xi) and (xii), we get
`2y=1`
`:.y=(1)/(2)`
Substituting value of b in equ. (x), we get
`c=b=(1)(2)`
From eq. (xii)
`a=-b-2c-1`
Substituting values of b and c we get
`x=-(1)/(2)-2((1)/(2))-1=-(5)/(2)`
Putting values of a,b and c in equ. (ix) we get
`T=kc^((5)/(2))h^((1)/(2))G^((1)/(2))`
`T=ksqrt((hG)/(c^(5)))`