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.

We know that, dimensions of $(h)=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]$ (From $\left.\mathrm{E}=h f\right]$ Dimensions of $(c)=\left[\mathrm{L}^{1} \mathrm{~T}^{-1}\right] \quad(c$ is velocity $)$

Dimensions of gravitational constant

$(\mathrm{G})=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right] \quad\left(\text { From } \mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}\right)$

$(i)$ Let $\mathrm{m} \propto c^{a} h^{b} \mathrm{G}^{c}$

$\Rightarrow \mathrm{m}=k c^{a} h^{b} \mathrm{G}^{c}$

where, $k$ is a dimensionless constant of proportionality. Substituting dimensions of each term in Eq.$ (i)$, we get

$\left[\mathrm{ML}^{0} \mathrm{~T}^{0}\right] =\left[\mathrm{LT}^{-1}\right]^{x} \times\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]^{y}\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]^{z}$

$=\left[\mathrm{M}^{b-c} \mathrm{~L}^{a+2 b+3 c} \mathrm{~T}^{-a-b-2 c}\right]$

Comparing powers of same terms on both sides, we get

$b-c=1\ldots \text { (ii) }$

$a+2 b+3 c=0\ldots\text { (iii) }$

$-a-b-2 c=0\ldots\text { (iv) }$

$\ldots \text { (ii) }$

$\ldots \text { (iii) }$

Adding Eqs. $(ii)$, $(iii)$ and $(iv)$, we get

$2 b=1 \Rightarrow b=\frac{1}{2}$

Substituting value of $\mathrm{b}$ in eq. $(ii)$, we get

$c=-\frac{1}{2}$

From eq. $(iv)$,

$a=-b-2 c$

Substituting values of $b$ and $c$, we get

$a=-\frac{1}{2}-2\left(-\frac{1}{2}\right)=\frac{1}{2}$