1. We are asked to simplify the radical expressions given in problem 8.a: $\sqrt{338a^3b^3}$. 2. Recall the rule for simplifying square roots: $\sqrt{x^2} = |x|$ and $\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$. We will use these to simplify each part. 3. Start by factoring the radicand (the expression inside the square root): $$338a^3b^3 = 169 \times 2 \times a^2 \times a \times b^2 \times b$$ 4. Apply the square root to each factor: $$\sqrt{338a^3b^3} = \sqrt{169} \times \sqrt{2} \times \sqrt{a^2} \times \sqrt{a} \times \sqrt{b^2} \times \sqrt{b}$$ 5. Simplify the perfect squares: $$\sqrt{169} = 13, \quad \sqrt{a^2} = a, \quad \sqrt{b^2} = b$$ 6. Substitute back: $$13 \times \sqrt{2} \times a \times \sqrt{a} \times b \times \sqrt{b} = 13ab \sqrt{2ab}$$ 7. Final simplified form: $$\boxed{13ab \sqrt{2ab}}$$ This completes the simplification of 8.a.