Barium nitrate has the formula
This means that each formula unit of barium nitrate contains two nitrate ions. The number of nitrate ions in 1.563x10^30 is thus 2(1.563x10^30), or 3.126x10^30 individual nitrate ions.
When you're trying to solve a problem like this, write out the formula for the ionic compound. The barium ion, being in Group 2 of the periodic table, forms an ion with a charge of +2. Nitrate is a polyatomic ion. One way to know this is that it ends in -ate. Monatomic ions (those with just one element) end in -ide if they have a negative charge. You can find the nitrate ion, NO3-, on a table of polyatomic ions and see that its charge is -1. An ionic compound has a net charge of zero, so the formula will have to contain 2 nitrate ions to cancel out the +2 charge of the barium ion.