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

(N/A) Given expression is $P = E l^2 m^{-5} G^{-2}$.
Dimensions of the physical quantities are:
$E$ (Energy) = $[M^1 L^2 T^{-2}]$
$l$ (Angular momentum) = $[M^1 L^2 T^{-1}]$
$m$ (Mass) = $[M^1]$
$G$ (Gravitational constant) = $[M^{-1} L^3 T^{-2}]$
Substituting these dimensions into the expression for $P$:
$[P] = [M^1 L^2 T^{-2}] \times [M^1 L^2 T^{-1}]^2 \times [M^1]^{-5} \times [M^{-1} L^3 T^{-2}]^{-2}$
Expanding the powers:
$[P] = [M^1 L^2 T^{-2}] \times [M^2 L^4 T^{-2}] \times [M^{-5}] \times [M^2 L^{-6} T^4]$
Combining the exponents for each base:
For $M$: $1 + 2 - 5 + 2 = 0$
For $L$: $2 + 4 + 0 - 6 = 0$
For $T$: $-2 - 2 + 0 + 4 = 0$
Thus,$[P] = [M^0 L^0 T^0]$.
Therefore,$P$ is a dimensionless quantity.