1. **Problem d:** Simplify the expression $$\frac{3 - \sqrt{2}}{6 - \sqrt{5}}$$. 2. **Step 1:** Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator $$6 + \sqrt{5}$$. $$\frac{3 - \sqrt{2}}{6 - \sqrt{5}} \times \frac{6 + \sqrt{5}}{6 + \sqrt{5}} = \frac{(3 - \sqrt{2})(6 + \sqrt{5})}{(6)^2 - (\sqrt{5})^2}$$ 3. **Step 2:** Calculate denominator: $$36 - 5 = 31$$ 4. **Step 3:** Expand numerator: $$3 \times 6 + 3 \times \sqrt{5} - \sqrt{2} \times 6 - \sqrt{2} \times \sqrt{5} = 18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10}$$ 5. **Step 4:** Final simplified form: $$\frac{18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10}}{31}$$ --- 6. **Problem e:** Simplify the expression $$\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}}$$. 7. **Step 1:** Multiply numerator and denominator by the conjugate of the denominator $$\sqrt{11} - \sqrt{7}$$: $$\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} \times \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}} = \frac{(\sqrt{11} - \sqrt{7})^2}{(\sqrt{11})^2 - (\sqrt{7})^2}$$ 8. **Step 2:** Calculate denominator: $$11 - 7 = 4$$ 9. **Step 3:** Expand numerator: $$(\sqrt{11})^2 - 2\sqrt{11}\sqrt{7} + (\sqrt{7})^2 = 11 - 2\sqrt{77} + 7 = 18 - 2\sqrt{77}$$ 10. **Step 4:** Final simplified form: $$\frac{18 - 2\sqrt{77}}{4}$$ 11. **Step 5:** Simplify fraction by dividing numerator and denominator by 2: $$\frac{\cancel{2}(9 - \sqrt{77})}{\cancel{2} \times 2} = \frac{9 - \sqrt{77}}{2}$$ --- 12. **Problem f:** Simplify the expression $$\frac{2}{(3 - \sqrt{2})^2}$$. 13. **Step 1:** Expand denominator using formula $$(a - b)^2 = a^2 - 2ab + b^2$$: $$(3 - \sqrt{2})^2 = 3^2 - 2 \times 3 \times \sqrt{2} + (\sqrt{2})^2 = 9 - 6\sqrt{2} + 2 = 11 - 6\sqrt{2}$$ 14. **Step 2:** Expression becomes: $$\frac{2}{11 - 6\sqrt{2}}$$ 15. **Step 3:** Rationalize denominator by multiplying numerator and denominator by conjugate $$11 + 6\sqrt{2}$$: $$\frac{2}{11 - 6\sqrt{2}} \times \frac{11 + 6\sqrt{2}}{11 + 6\sqrt{2}} = \frac{2(11 + 6\sqrt{2})}{(11)^2 - (6\sqrt{2})^2}$$ 16. **Step 4:** Calculate denominator: $$121 - 36 \times 2 = 121 - 72 = 49$$ 17. **Step 5:** Final simplified form: $$\frac{22 + 12\sqrt{2}}{49}$$ --- 18. **Problem 4:** Given a cuboid with volume 2 m³ and edges $$5 - \sqrt{3}$$ cm, $$2 + \sqrt{3}$$ cm, and $$x$$ cm, find $$x$$ in simplified surd form with rationalized denominator. 19. **Step 1:** Volume formula for cuboid: $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$ 20. **Step 2:** Convert volume to cm³ (since edges are in cm): $$2 \text{ m}^3 = 2 \times 100^3 = 2,000,000 \text{ cm}^3$$ 21. **Step 3:** Set up equation: $$(5 - \sqrt{3})(2 + \sqrt{3})x = 2,000,000$$ 22. **Step 4:** Multiply the two surd expressions: $$(5)(2) + 5\sqrt{3} - 2\sqrt{3} - (\sqrt{3})^2 = 10 + 5\sqrt{3} - 2\sqrt{3} - 3 = 7 + 3\sqrt{3}$$ 23. **Step 5:** Solve for $$x$$: $$x = \frac{2,000,000}{7 + 3\sqrt{3}}$$ 24. **Step 6:** Rationalize denominator by multiplying numerator and denominator by conjugate $$7 - 3\sqrt{3}$$: $$x = \frac{2,000,000 (7 - 3\sqrt{3})}{(7)^2 - (3\sqrt{3})^2} = \frac{2,000,000 (7 - 3\sqrt{3})}{49 - 27} = \frac{2,000,000 (7 - 3\sqrt{3})}{22}$$ 25. **Step 7:** Simplify fraction: $$\frac{2,000,000}{22} = \frac{1,000,000}{11}$$ 26. **Step 8:** Final simplified expression for $$x$$: $$x = \frac{1,000,000}{11} (7 - 3\sqrt{3})$$ --- **Final answers:** d) $$\frac{18 + 3\sqrt{5} - 6\sqrt{2} - \sqrt{10}}{31}$$ e) $$\frac{9 - \sqrt{77}}{2}$$ f) $$\frac{22 + 12\sqrt{2}}{49}$$ 4) $$x = \frac{1,000,000}{11} (7 - 3\sqrt{3})$$ (cm)