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(D) Let the dimension of length be $L = G^x c^y h^z$.The dimensions are:$G = [M^{-1} L^3 T^{-2}]$$c = [L T^{-1}]$$h = [M L^2 T^{-1}]$Substituting thes

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$(b)$ and $(c)$ both

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(D) Let the dimension of length be $L = G^x c^y h^z$.The dimensions are:$G = [M^{-1} L^3 T^{-2}]$$c = [L T^{-1}]$$h = [M L^2 T^{-1}]$Substituting these into the equation:$[M^0 L^1 T^0] = [M^{-1} L^3 T^{-2}]^x [L T^{-1}]^y [M L^2 T^{-1}]^z$$[M^0 L^1 T^0] = M^{-x+z} L^{3x+y+2z} T^{-2x-y-z}$Comparing the powers on both sides:$1$) $-x + z = 0 \implies x = z$$2$) $3x + y + 2z = 1$$3$) $-2x - y - z = 0 \implies y = -2x - z$Substitute $x=z$ into $(3)$: $y = -2x - x = -3x$.Substitute $x=z$ and $y=-3x$ into $(2)$:$3x + (-3x) + 2x = 1$$2x = 1 \implies x = 1/2$.Since $x=z$,$z = 1/2$.Since $y = -3x$,$y = -3/2$.Thus,$x = 1/2, y = -3/2, z = 1/2$. Both options $(b)$ and $(c)$ are correct.

If the dimensions of length are expressed as ${G^x}{c^y}{h^z}$; where $G, c$ and $h$ are the universal gravitational constant,speed of light,and Planck's constant respectively,then:

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