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

Explanation

Here, `c=[LT^-1],h=[ML^2T^-1],`
`G=[M^-1L^3T^-2]`
`becauseE=hv,h=E/v,G=((Fd^2)/(m_1m_2))`
Let `m=c^xh^yG^zto(1)`
`implies[M^1L^0T^0]=(LT^-1)^x(ML^2T^-1)^y(M^-1L^3T^-2)^z`
`implies[M^1L^0T^0]=M^(y-z)L^(x+2y+2z)T^(-x-y-2z)`
Applying the principle of homegenity of dimensions, we get
`y-z=1to(2),x+2y+3z=0to(3),`
`-x-y-2z=0to(4)`
Adding eq.(2), eq.(3) and eq.(4).
`2y=1impliesy=1/2`
`therefore` From eq.(2) `z=y-1=1/2-1=(-1)/2`
From eq.(4) `x=-y-2z=(-1)/2+1=1/2`
Substituting the values of x,y & z in eq.(1) , we get
`m=c^(1//2)h^(1//2)G^(-1//2)impliesm=sqrt((ch)/G)`
Proceeding as above we can show that
`L=sqrt((hG)/c^3)andT=sqrt((hG)/c^5)`