In the expression $P = El^2m^{-5}G^{-2}$, $E$, $l$, $m$ and $G$ denote energy, mass, angular momentum and gravitational constant respectively. Show that $P$ is a dimensionless quantity.

Given, expression is, $\quad P=E L^{2} m^{-5} G^{-2}$

where $\mathrm{E}$ is energy

$[\mathrm{E}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]$

$\mathrm{m}$ is mass

$[\mathrm{m}]=\left[\mathrm{M}^{1}\right]$

$\mathrm{L}$ is angular momentum $[\mathrm{L}]=\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]$

$\mathrm{G}$ is gravitational constant $[\mathrm{G}]=\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]$

Substituting dimensions of each term in the given expression,

$[\mathrm{P}] =\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right] \times\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]^{2} \times\left[\mathrm{M}^{1}\right]^{-5} \times\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]^{-2}$

$=\left[\mathrm{M}^{1+2-5+2} \mathrm{~L}^{2+4-6} \mathrm{~T}^{-2-2+4}\right]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]$

Hence, $P$ is a dimensionless quantity.