At a given temperature, if the solublity product and the solublity of a sparingly soluble salt `M_(2)X_(3)` be `K_(sp) and S`.
respectively, then prove that `S=((K_(sp))/(108))^(1//(5))`.

Explanation

In its saturated solution, `M_(2)X_(3)` ionises partially, forming `M^(3+)(aq) and X^(2-)(aq)` ions. Eventually the following equilibriu is established.
`M_(2)X_(3)(s)hArr 2M^(3+)(aq)+3X^(2-)(aq)`
if the solubility of `M_(2)X_(3)` in its saturated solution is S `mol*L^(-1)`, then in the solution `[M^(3+)]=2S" "mol*L^(-1)` and `[X^(2-)]=3S" "mol*L^(-1)`
Therefore, the solubility product of `M_(2)X_(3), K_(sp)=[M^(3+)]^(2)[X^(2-)]^(3)=(2S)^(2)(3S)^(2)=108S^(5)`
`therefore S=sqrt((K_(sp))/(108))^((1)/(5))`.