A physical quantity X is related to the permittivity of free space ₀ , Planck's constant h , and the speed of light c by the relation X ₀^ h^ c^ . If the values of the exponents are = -1/2 , = 1/2 , and = 1/2 , then the quantity X represents:

Step-by-step solution

First, write the dimensional formulas for the given fundamental constants: Permittivity of free space [ ₀] = [M⁻¹ L⁻³ T^4 A^2] Planck's constant [h] = [M L^2 T⁻¹] Speed of light [c] = [L T⁻¹] Substitute the given exponents into the relation X ₀^ -1/2 h^ 1/2 c^ 1/2 : [X] = [M⁻¹ L⁻³ T^4 A^2]^ -1/2 [M L^2 T⁻¹]^ 1/2 [L T⁻¹]^ 1/2 Simplify the powers for each dimension: For M : (-1)(-1/2) + (1)(1/2) + (0) = 1/2 + 1/2 = 1 For L : (-3)(-1/2) + (2)(1/2) + (1)(1/2) = 3/2 + 1 + 1/2 = 3 For T : (4)(-1/2) + (-1)(1/2) + (-1)(1/2