In a new system of units, Planck's constant ( h ), the speed of light in vacuum ( c ), and the universal gravitational constant ( G ) are chosen as fundamental quantities. If the dimensional formula of an unknown physical quantity X in this system is [h⁻¹ c^7 G⁻²] , then X represents:

Step-by-step solution

The dimensional formulas of the given fundamental constants in terms of mass ( M ), length ( L ), and time ( T ) are: [h] = [M L^2 T⁻¹] [c] = [L T⁻¹] [G] = [M⁻¹ L^3 T⁻²] The dimensional formula of the unknown quantity X is given as [h⁻¹ c^7 G⁻²] . Substituting the dimensions of h , c , and G : [X] = [M L^2 T⁻¹]⁻¹ [L T⁻¹]^7 [M⁻¹ L^3 T⁻²]⁻² Expanding the powers: [X] = (M⁻¹ L⁻² T^1) (L^7 T⁻⁷) (M^2 L⁻⁶ T^4) Combining the terms for M , L , and T : [X] = M^ -1 + 2 L^ -2 + 7 - 6 T^ 1 - 7 + 4 [X] = M^1 L⁻¹ T⁻² The dimensio