If Planck's constant (h) , speed of light in vacuum (c) and Newton's gravitational constant (G) are taken as fundamental quantities, which of the following physical quantities has the dimensional formula [h^ 1/2 c^ -5/2 G^ 1/2 ] ?

Step-by-step solution

The standard dimensional formulas of the given constants are: Planck's constant, h = [ML^2T⁻¹] Speed of light, c = [LT⁻¹] Gravitational constant, G = [M⁻¹L^3T⁻²] Let us evaluate the dimensions of the given expression [h^ 1/2 c^ -5/2 G^ 1/2 ] : First, find the product hG : hG = [ML^2T⁻¹] [M⁻¹L^3T⁻²] = [L^5T⁻³] Now, raise it to the power of 1/2 : (hG)^ 1/2 = [L^ 5/2 T^ -3/2 ] Next, evaluate c^ -5/2 : c^ -5/2 = [LT⁻¹]^ -5/2 = [L^ -5/2 T^ 5/2 ] Multiplying these terms together: [h^ 1/2 c^ -5/2 G^ 1/2 ] = [L^ 5/2 T^ -3/