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Chemistry Level 3

The solubility product of calcium phosphate \( (\ce{Ca3(PO4)2}\) is \( \beta \) at \(25^\circ \text{ C}\). Let \(S_{1}\) be the solubility of calcium phosphate in water at \(25^\circ \text{ C}\) and \(S_{2}\) be the solubility of calcium phosphate in a certain solution of calcium nitrate at the same temperature (where \(S_{1}\) and \(S_{2}\) in \(\text{mol/L}\)).

Given that the calcium nitrate solution is isotonic to a \(4.5\text{ mol/L}\) sodium chloride solution, the ratio \( \dfrac{S_{1}}{S_{2}}\) can be expressed as \[ \large 2^{a}3^{b} \beta ^{-c},\] where \(a\), \(b\) and \(c\) are positive, rational numbers. What's the value of \( \left(\dfrac{b+c}{a}\right)^{4} \)?

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