1. **Problem Statement:** Simplify the expressions: d) $\frac{3 - \sqrt{2}}{6 - \sqrt{5}}$ e) $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ f) $\frac{2}{(3 - \sqrt{2})^2}$ --- 2. **Rationalizing the denominator:** To simplify expressions with square roots in the denominator, multiply numerator and denominator by the conjugate of the denominator or expand and simplify. --- ### d) Simplify $\frac{3 - \sqrt{2}}{6 - \sqrt{5}}$ - Multiply numerator and denominator by the conjugate of the denominator $6 + \sqrt{5}$: $$\frac{3 - \sqrt{2}}{6 - \sqrt{5}} \times \frac{6 + \sqrt{5}}{6 + \sqrt{5}} = \frac{(3 - \sqrt{2})(6 + \sqrt{5})}{(6 - \sqrt{5})(6 + \sqrt{5})}$$ - Calculate denominator using difference of squares: $$6^2 - (\sqrt{5})^2 = 36 - 5 = 31$$ - Expand numerator: $$ (3)(6) + 3\sqrt{5} - 6\sqrt{2} - \sqrt{2}\sqrt{5} = 18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10} $$ - So expression is: $$\frac{18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10}}{31}$$ --- ### e) Simplify $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$ - Multiply numerator and denominator by conjugate $\sqrt{11} - \sqrt{7}$: $$\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} \times \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}} = \frac{(\sqrt{11} - \sqrt{7})^2}{(\sqrt{11})^2 - (\sqrt{7})^2}$$ - Calculate denominator: $$11 - 7 = 4$$ - Expand numerator: $$ (\sqrt{11})^2 - 2\sqrt{11}\sqrt{7} + (\sqrt{7})^2 = 11 - 2\sqrt{77} + 7 = 18 - 2\sqrt{77} $$ - So expression is: $$\frac{18 - 2\sqrt{77}}{4}$$ - Simplify by dividing numerator and denominator by 2: $$\frac{\cancel{2}(9 - \sqrt{77})}{\cancel{2} \times 2} = \frac{9 - \sqrt{77}}{2}$$ --- ### f) Simplify $\frac{2}{(3 - \sqrt{2})^2}$ - Expand denominator: $$ (3 - \sqrt{2})^2 = 3^2 - 2 \times 3 \times \sqrt{2} + (\sqrt{2})^2 = 9 - 6\sqrt{2} + 2 = 11 - 6\sqrt{2} $$ - So expression is: $$\frac{2}{11 - 6\sqrt{2}}$$ - Multiply numerator and denominator by conjugate $11 + 6\sqrt{2}$: $$\frac{2}{11 - 6\sqrt{2}} \times \frac{11 + 6\sqrt{2}}{11 + 6\sqrt{2}} = \frac{2(11 + 6\sqrt{2})}{(11)^2 - (6\sqrt{2})^2}$$ - Calculate denominator: $$121 - 36 \times 2 = 121 - 72 = 49$$ - So expression is: $$\frac{22 + 12\sqrt{2}}{49}$$ --- **Final answers:** d) $\frac{18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10}}{31}$ e) $\frac{9 - \sqrt{77}}{2}$ f) $\frac{22 + 12\sqrt{2}}{49}$