There are multiple chemical reactions happening here. Let's break them down one by one:

MgCO3 + HNO3 = Magnesium carbonate reacts with nitric acid to form magnesium nitrate and water. The balanced equation is:
MgCO3 + 2 HNO3 = Mg(NO3)2 + CO2 + H2O

NaOH + FeCl3 = Sodium hydroxide reacts with iron(III) chloride to form sodium chloride and iron(III) hydroxide. The balanced equation is:
3 NaOH + FeCl3 = 3 NaCl + Fe(OH)3

Ba(OH)2 + HCl = Barium hydroxide reacts with hydrochloric acid to form barium chloride and water. The balanced equation is:
Ba(OH)2 + 2 HCl = BaCl2 + 2 H2O

These are the balanced chemical equations for the reactions mentioned.