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Can you give me an example of a compound that can make Ba3(PO4)2 even more insoluble than it already is (and can you tell me why, just for my own understanding)?

You use the common ion effect which is just an extension of the Le Chatelier's Principle. The solubility of Ba3(PO4)2 can be written as follows:
Ba3(PO4)2(s) + H2O ==> 3Ba^+2(aq) + 2PO4^-3(aq)

The compound does have a small solubility. Now, if we add EITHER Ba^+ from a soluble compound, such as BaCl2, or PO4^-3 from a soluble compound, such as Na3PO4, (those are "common ions"), then the solubility equation written above will shift to the left. Of course, that makes Ba3(PO4)2 even more insoluble.
I hope this clears up any confusion you may be experiencing. Thanks for using Jiskha.

Thank you so much - that's an awesome explanation and clears things up so well!! :)

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