1. Write the balanced chemical equation:

Ba(OH)₂(aq) + 2HClO₄(aq) → Ba(ClO₄)₂(aq) + 2H₂O(l)

2. Calculate the moles of perchloric acid (HClO₄):

* Molarity (M) = moles (mol) / volume (L)

* Moles of HClO₄ = Molarity × volume = 0.597 M × 0.0147 L = 0.00876 mol

3. Determine the mole ratio from the balanced equation:

* The balanced equation shows that 1 mole of Ba(OH)₂ reacts with 2 moles of HClO₄.

4. Calculate the moles of barium hydroxide (Ba(OH)₂):

* Moles of Ba(OH)₂ = (moles of HClO₄) / 2 = 0.00876 mol / 2 = 0.00438 mol

5. Calculate the mass of barium hydroxide:

* Mass (g) = moles (mol) × molar mass (g/mol)

* Molar mass of Ba(OH)₂ = 137.33 g/mol + 2(17.01 g/mol) = 171.35 g/mol

* Mass of Ba(OH)₂ = 0.00438 mol × 171.35 g/mol = 0.750 g

Therefore, you need 0.750 grams of solid barium hydroxide to exactly neutralize 14.7 mL of a 0.597 M perchloric acid solution.