If the velocity of light c, Planks 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.

If the velocity of light c, Planks constant, h and the gravitational c

1.	If the velocity of light c, Planks 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.
Answer» Solution :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)`