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I don't know how to obtain the "exact charge of the ions" that will be formed by each atom mentioned. I also have to figure out whether they are positive or negative.

Hydrogen, for example, can form the ion:

$$\ce{H -> H+ + e-}$$

Now if I were to try to get the exact charge for say, boron using only the periodic table, how would I do so?

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Each atom tends to imitate the noble gases which have a full valence shell. Looking at your table you can see that boron is 3 spaces above helium, but 5 spaces behind neon. Boron would "prefer" to have a configuration like the noble gases, and because losing 3 electrons seems easier than gaining 5, boron, as boring as he is, decides to do the obvious thing and three electrons walk the plank... Now he has 5 protons and only 2 electrons therefore imitating helium's electron configuration. This means boron has a charge of $+3$.

Oxidation states are common on most periodic tables, which show you the result of less than obvious behavior of atoms when this trend does not always hold. In the case of hydrogen, it is common than it will bond with another hydrogen to imitate helium, otherwise it will tend to form a $+1$ ion.

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