1. **Problem 9:** Simplify $-\sqrt{6}n(\sqrt{6} + 2)$. 2. Use the distributive property: $-\sqrt{6}n \times \sqrt{6} + (-\sqrt{6}n) \times 2$. 3. Calculate each term: $$-\sqrt{6}n \times \sqrt{6} = -n \times \sqrt{6} \times \sqrt{6} = -n \times \sqrt{36} = -n \times 6 = -6n$$ $$-\sqrt{6}n \times 2 = -2\sqrt{6}n$$ 4. Combine terms: $$-6n - 2\sqrt{6}n$$ This is the simplified form; the expression $-\sqrt{36}n$ is correct but $-xn$ is incorrect. --- 1. **Problem 11:** Simplify $(2\sqrt{3} + 4)(4\sqrt{3} - 4)$. 2. Use the distributive property (FOIL): $$2\sqrt{3} \times 4\sqrt{3} + 2\sqrt{3} \times (-4) + 4 \times 4\sqrt{3} + 4 \times (-4)$$ 3. Calculate each term: $$2 \times 4 \times \sqrt{3} \times \sqrt{3} = 8 \times 3 = 24$$ $$2\sqrt{3} \times (-4) = -8\sqrt{3}$$ $$4 \times 4\sqrt{3} = 16\sqrt{3}$$ $$4 \times (-4) = -16$$ 4. Combine like terms: $$24 + (-8\sqrt{3} + 16\sqrt{3}) - 16 = 24 + 8\sqrt{3} - 16$$ 5. Simplify constants: $$24 - 16 = 8$$ 6. Final simplified expression: $$8 + 8\sqrt{3}$$ --- 1. **Problem 13:** Simplify the expression $\frac{5}{2 - \sqrt{2}}$. 2. Rationalize the denominator by multiplying numerator and denominator by the conjugate $2 + \sqrt{2}$: $$\frac{5}{2 - \sqrt{2}} \times \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{5(2 + \sqrt{2})}{(2)^2 - (\sqrt{2})^2}$$ 3. Calculate denominator: $$4 - 2 = 2$$ 4. Expand numerator: $$5 \times 2 + 5 \times \sqrt{2} = 10 + 5\sqrt{2}$$ 5. Final simplified expression: $$\frac{10 + 5\sqrt{2}}{2} = 5 + \frac{5}{2}\sqrt{2}$$ **Final answers:** - Problem 9: $-6n - 2\sqrt{6}n$ - Problem 11: $8 + 8\sqrt{3}$ - Problem 13: $5 + \frac{5}{2}\sqrt{2}$