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(A) Let the distance $d$ be given by $d = k \rho^a S^b f^c$,where $k$ is a dimensionless constant.The dimensions are:$[d] = [L]$$[\rho] = [M L^{-3}]$$

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$3$

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(A) Let the distance $d$ be given by $d = k \rho^a S^b f^c$,where $k$ is a dimensionless constant.The dimensions are:$[d] = [L]$$[\rho] = [M L^{-3}]$$[S] = [Power/Area] = [M L^2 T^{-3} / L^2] = [M T^{-3}]$$[f] = [T^{-1}]$Substituting these into the equation:$[L]^1 = [M L^{-3}]^a [M T^{-3}]^b [T^{-1}]^c$$[L]^1 = M^{a+b} L^{-3a} T^{-3b-c}$Comparing the powers of $M, L, T$ on both sides:For $M: a + b = 0 \Rightarrow a = -b$For $L: -3a = 1 \Rightarrow a = -1/3$Thus,$b = 1/3$For $T: -3b - c = 0 \Rightarrow c = -3b = -3(1/3) = -1$Since $d \propto S^b$ and $b = 1/3$,we have $d \propto S^{1/3}$.Comparing this with $d \propto S^{1/n}$,we get $n = 3$.

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