$\frac{3 + \sqrt{3}}{2\sqrt{3}} = \frac{(\sqrt{3})^2 + \sqrt{3}}{2\sqrt{3}} = \frac{\sqrt{3}(\sqrt{3} + 1)}{2\sqrt{3}} = \frac{\sqrt{3} + 1}{2}$

$\frac{5 - \sqrt{5}}{\sqrt{10} - 5\sqrt{2}} = \frac{(\sqrt{5})^2 - \sqrt{5}}{\sqrt{5 * 2} - 5\sqrt{2}} = \frac{\sqrt{5}(\sqrt{5} - 1)}{\sqrt{2} * \sqrt{5} - 5\sqrt{2}} = \frac{\sqrt{5}(\sqrt{5} - 1)}{\sqrt{2}(\sqrt{5} - 5)} = \frac{\sqrt{5}(\sqrt{5} - 1)}{\sqrt{2}(\sqrt{5} - (\sqrt{5})^2)} = \frac{\sqrt{5}(\sqrt{5} - 1)}{\sqrt{2} * \sqrt{5}(1 - \sqrt{5})} = -\frac{\sqrt{5} - 1}{\sqrt{2}(\sqrt{5} - 1)} = -\frac{1}{\sqrt{2}} = -\frac{1 * \sqrt{2}}{\sqrt{2} * \sqrt{2}} = -\frac{\sqrt{2}}{2}$

$\frac{2 - \sqrt{6}}{\sqrt{6} - 3} = \frac{(\sqrt{2})^2 - \sqrt{2} * \sqrt{3}}{\sqrt{2} * \sqrt{3} - (\sqrt{3})^2} = \frac{\sqrt{2}(\sqrt{2} - \sqrt{3})}{\sqrt{3}(\sqrt{2} - \sqrt{3})} = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2} * \sqrt{3}}{\sqrt{3} * \sqrt{3}} = \frac{\sqrt{6}}{3}$

$\frac{4a - 2}{2\sqrt{a} + \sqrt{2}} = \frac{2(2a - 1)}{(\sqrt{2})^2 * \sqrt{a} + \sqrt{2}} = \frac{(\sqrt{2})^2((\sqrt{2a})^2 - 1^2)}{\sqrt{2}(\sqrt{2a} + 1)} = \frac{\sqrt{2}(\sqrt{2a} - 1)(\sqrt{2a} + 1)}{\sqrt{2a} + 1} = \sqrt{2}(\sqrt{2a} - 1)$

$\frac{9a - b^2}{9a + 6b\sqrt{a} + b^2} = \frac{(3\sqrt{a})^2 - b^2}{(3\sqrt{a})^2 + 2 * 3\sqrt{a} * b + b^2} = \frac{(3\sqrt{a} - b)(3\sqrt{a} + b)}{(3\sqrt{a} + b)^2} = \frac{3\sqrt{a} - b}{3\sqrt{a} + b}$

$\frac{a\sqrt{a} - 8}{a + 2\sqrt{a} + 4} = \frac{(\sqrt{a})^2 * \sqrt{a} - 2^3}{a + 2\sqrt{a} + 4} = \frac{(\sqrt{a})^3 - 2^3}{a + 2\sqrt{a} + 4} = \frac{(\sqrt{a} - 2)((\sqrt{a})^2 + 2\sqrt{a} + 2^2)}{a + 2\sqrt{a} + 4} = \frac{(\sqrt{a} - 2)(a + 2\sqrt{a} + 4)}{a + 2\sqrt{a} + 4} = \sqrt{a} - 2$