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(C) Let the dimension of length $l$ be expressed as $l \propto h^p c^q G^r$.Substituting the dimensions of each constant:$[L] = [ML^2T^{-1}]^p [LT^{-1

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$\frac{\sqrt{hG}}{c^{3/2}}$

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(C) Let the dimension of length $l$ be expressed as $l \propto h^p c^q G^r$.Substituting the dimensions of each constant:$[L] = [ML^2T^{-1}]^p [LT^{-1}]^q [M^{-1}L^3T^{-2}]^r$$[M^0 L^1 T^0] = M^{p-r} L^{2p+q+3r} T^{-p-q-2r}$Comparing the powers of $M, L,$ and $T$ on both sides:$p - r = 0 \implies p = r$$2p + q + 3r = 1$$-p - q - 2r = 0$Substituting $p = r$ into the third equation: $-r - q - 2r = 0 \implies q = -3r$.Substituting $p = r$ and $q = -3r$ into the second equation: $2r - 3r + 3r = 1 \implies 2r = 1 \implies r = 1/2$.Thus,$p = 1/2$ and $q = -3/2$.Therefore,$l \propto h^{1/2} c^{-3/2} G^{1/2} = \sqrt{\frac{hG}{c^3}} = \frac{\sqrt{hG}}{c^{3/2}}$.

Planck's constant $(h)$,speed of light in vacuum $(c)$,and Newton's gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length?

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