Mn: 3+
Cl: 7+
O: 2-
Explanation:
1) Compound given: Mn [ClO₄]₃
2) Initially you only know the oxidation state of O, since it is always 2-, except when it form peroxides, which is not the case.
3) So, you do not know the oxidation states neither of the Mn nor of the Cl, and you need some more information.
You might start from the ion [ClO₄] but you do not know its charge.
This ion comes from one of the oxoacids formed by Cl. Those are four different acids. These are them:
i) Oxidation state 1+: Cl₂O + H₂O → H₂Cl₂O₂ = HClO ⇒ ion ClO⁻
ii) Oxidation state 3+: Cl₂O₃ + H₂O → H₂Cl₂O₄ = HClO₂ ⇒ ion ClO₂⁻
iii) Oxidation state 5+: Cl₂O5 + H₂O → H₂Cl₂O₆ = HClO₃ ⇒ ion ClO₃⁻
iv) Oxidation state 7+: Cl₂O₇ + H₂O → H₂Cl₂O₈ = HClO₄ ⇒ ion ClO₄⁻
Finally, we have that our ion is ClO₄⁻ and the oxidation state of Cl is 7+.
4) Now you just have to find the oxidation state of Mn, for which you make a balance of charges:
Mn [ClO₄]₃
Since, the ion ClO₄⁻ has 1 negative charge, and there are 3 ions the total negative charge is 3-. Since the compound is neutral, you conclude that Mn has oxidation state 3+.
That according to this balance: 1(3+) + 3(1-) + 3 - 3 = 0.
5) Summarizing, the oxidation states are:
Mn: 3+
Cl: 7+
O: 2-
