If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express length in terms of dimensions of these quantities.
If velocity of light $c$, Planck’s constant $h$ and gravitational constant $G$ are taken as fundamental quantities, then express length in terms of dimensions of these quantities.
Let $l \propto c^{x} y^{y} G^{z} ; 1=k c^{x} h^{y} G^{z}$
where $k$ is a dimensionless constant and $x , y$ and $z$ are the exponents.
Equating dimensions on both sides, we get
${\left[ M ^{0} LT ^{0}\right]=\left[ LT ^{-1}\right]^{ x }\left[ ML ^{2} T ^{-1}\right]^{y}\left[ M ^{-1} L ^{3} T ^{-2}\right]^{z}}$
$=\left[ M ^{y-z} L ^{ x +2 y +3 z } T ^{- x - y -2 z }\right]$
Applying the principle of homogeneity of dimensions, we get
$y-z=0$
$x+2 y+3 z=1$
$-x-y-2 z=0$
$x =\frac{-3}{2}, y =\frac{1}{2} z =\frac{1}{2}$
$l=\sqrt{\frac{ hG }{ c ^{3}}}$
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