In the expression `P =E L^2 m^(-5) G^(-2),` E, m, L and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.

Explanation

Taking dimensional formula
energy `(E ) = ML^2 T^(-2)`
mass `(M) = ML^0 T^0`
Angular Momentum `(L) = ML^2 T^(-1)`
Universal Gravitational constant `(G) = M^(-1) L^(3) T^(-2)`
Substituting in `(EL^2)/(M^5 G^2) `, we get
`((ML^2T^(-2))(ML^2T^(-1))^2)/((ML^0 T^0)^5 (M^(-1)L^3 T^(-2))^2) = (M^(1 + 2) L^(2+4) T^(-2-2))/(M^(5-2)L^(0+6)T^(0-4))`
= A dimensionless quantity.