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.

(A) We know that the dimensions are:
Dimensions of $(h) = [M^1 L^2 T^{-1}]$
Dimensions of $(c) = [L^1 T^{-1}]$
Dimensions of $(G) = [M^{-1} L^3 T^{-2}]$
To express mass $(M)$,length $(L)$,and time $(T)$ in terms of $c, h, G$:
Let $M = k c^a h^b G^d$. Substituting dimensions:
$[M^1 L^0 T^0] = [L T^{-1}]^a [M L^2 T^{-1}]^b [M^{-1} L^3 T^{-2}]^d$
Equating powers:
$M: b - d = 1$
$L: a + 2b + 3d = 0$
$T: -a - b - 2d = 0$
Solving these equations gives $b = 1/2, d = -1/2, a = 1/2$.
Thus,$M = k \sqrt{\frac{hc}{G}}$.
Similarly,for length $(L)$ and time $(T)$,we solve for exponents to get:
$L = k \sqrt{\frac{hG}{c^3}}$ and $T = k \sqrt{\frac{hG}{c^5}}$.