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Question

(C) Let mass $M$ be expressed as $M \propto c^x G^y h^z$.Substituting the dimensions:$[M] = M^1 L^0 T^0$$[c] = L T^{-1}$$[G] = M^{-1} L^3 T^{-2}$$[h]

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

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(C) Let mass $M$ be expressed as $M \propto c^x G^y h^z$.Substituting the dimensions:$[M] = M^1 L^0 T^0$$[c] = L T^{-1}$$[G] = M^{-1} L^3 T^{-2}$$[h] = M L^2 T^{-1}$Thus,$M^1 L^0 T^0 = (L T^{-1})^x (M^{-1} L^3 T^{-2})^y (M L^2 T^{-1})^z$$M^1 L^0 T^0 = M^{-y+z} L^{x+3y+2z} T^{-x-2y-z}$Comparing the powers of $M, L, T$ on both sides:$1$) $-y + z = 1$$2$) $x + 3y + 2z = 0$$3$) $-x - 2y - z = 0$Adding $(2)$ and $(3)$: $y + z = 0 \implies y = -z$.Substituting $y = -z$ into $(1)$: $-(-z) + z = 1 \implies 2z = 1 \implies z = 1/2$.Then $y = -1/2$.Substituting $y$ and $z$ into $(3)$: $-x - 2(-1/2) - (1/2) = 0 \implies -x + 1 - 1/2 = 0 \implies x = 1/2$.Therefore,the dimensions of mass are $c^{1/2} G^{-1/2} h^{1/2}$.

If the velocity of light $(c)$,gravitational constant $(G)$,and Planck's constant $(h)$ are chosen as fundamental units,then the dimensions of mass in this new system are:

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