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Polarizing power is the ability to polarize another atom. So cations have high polarizing power because they have higher electrostatic pulls to neighbouring from their positive charge, and anions are highly polarizable since they have an abundance of electrons.
A good way to remember this is that these two characteristics will come with a pair of atoms, one being an anion and one being a cation. When you think of polarizing power think of how having this power creates action, so a cation is creating action by moving this electrons away from their original atom. If the cation has the polarizing power, then the anion has a polarizability, because it can be affected by the "power" of the cation.
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