# So many solutions!

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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