The speed of light (c),gravitational constant | vedclass

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(A) Let time $T \propto c^{x} G^{y} h^{z}$.$\Rightarrow T = k c^{x} G^{y} h^{z}$.Taking dimensions on both sides: $[M^{0} L^{0} T^{1}] = [L T^{-1}]^{x

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$G^{1/2} h^{1/2} c^{-5/2}$

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(A) Let time $T \propto c^{x} G^{y} h^{z}$.$\Rightarrow T = k c^{x} G^{y} h^{z}$.Taking dimensions on both sides: $[M^{0} L^{0} T^{1}] = [L T^{-1}]^{x} [M^{-1} L^{3} T^{-2}]^{y} [M L^{2} T^{-1}]^{z}$.$[M^{0} L^{0} T^{1}] = [M^{-y+z} L^{x+3y+2z} T^{-x-2y-z}]$.Equating powers of $M, L, T$ on both sides:$-y + z = 0 \implies z = y \quad \dots(1)$$x + 3y + 2z = 0 \quad \dots(2)$$-x - 2y - z = 1 \quad \dots(3)$Adding $(2)$ and $(3)$: $(x + 3y + 2z) + (-x - 2y - z) = 0 + 1 \implies y + z = 1$.Since $z = y$,we have $2y = 1 \implies y = 1/2$.Thus,$z = 1/2$.Substituting into $(2)$: $x + 3(1/2) + 2(1/2) = 0 \implies x + 3/2 + 1 = 0 \implies x = -5/2$.Therefore,$[T] = [G^{1/2} h^{1/2} c^{-5/2}]$.

The speed of light $(c)$,gravitational constant $(G)$,and Planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be

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