$\frac{\sqrt{a} - 3}{\sqrt{a} + 1} - \frac{\sqrt{a} - 4}{\sqrt{a}} = \frac{\sqrt{a}(\sqrt{a} - 3) - (\sqrt{a} + 1)(\sqrt{a} - 4)}{\sqrt{a}(\sqrt{a} + 1)} = \frac{a - 3\sqrt{a} - (a + \sqrt{a} - 4\sqrt{a} - 4)}{\sqrt{a}(\sqrt{a} + 1)} = \frac{a - 3\sqrt{a} - a - \sqrt{a} + 4\sqrt{a} + 4}{\sqrt{a}(\sqrt{a} + 1)} = \frac{4}{a + \sqrt{a}}$

$\frac{\sqrt{a} + 1}{a - \sqrt{ab}} - \frac{\sqrt{b} + 1}{\sqrt{ab} - b} = \frac{\sqrt{a} + 1}{\sqrt{a}(\sqrt{a} - \sqrt{b})} - \frac{\sqrt{b} + 1}{\sqrt{b}(\sqrt{a} - \sqrt{b})} = \frac{\sqrt{b}(\sqrt{a} + 1) - \sqrt{a}(\sqrt{b} + 1)}{\sqrt{ab}(\sqrt{a} - \sqrt{b})} = \frac{\sqrt{ab} + \sqrt{b} - \sqrt{ab} - \sqrt{a}}{\sqrt{ab}(\sqrt{a} - \sqrt{b})} = \frac{\sqrt{b} - \sqrt{a}}{\sqrt{ab}(\sqrt{a} - \sqrt{b})} = -\frac{\sqrt{a} - \sqrt{b}}{\sqrt{ab}(\sqrt{a} - \sqrt{b})} = -\frac{1}{\sqrt{ab}} = -\sqrt{\frac{1}{ab}}$

$\frac{\sqrt{x}}{y - 2\sqrt{y}} : \frac{\sqrt{x}}{3\sqrt{y} - 6} = \frac{\sqrt{x}}{\sqrt{y}(\sqrt{y} - 2)} : \frac{\sqrt{x}}{3(\sqrt{y} - 2)} = \frac{\sqrt{x}}{\sqrt{y}(\sqrt{y} - 2)} * \frac{3(\sqrt{y} - 2)}{\sqrt{x}} = \frac{1}{\sqrt{y}} * \frac{3}{1} = \frac{3}{\sqrt{y}}$

$\frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} : (\frac{\sqrt{m} + \sqrt{n}}{\sqrt{n}} + \frac{\sqrt{n}}{\sqrt{m} - \sqrt{n}}) = \frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} : \frac{(\sqrt{m} - \sqrt{n})(\sqrt{m} + \sqrt{n}) + (\sqrt{n})^2}{\sqrt{n}(\sqrt{m} - \sqrt{n})} = \frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} : \frac{m - n + n}{\sqrt{n}(\sqrt{m} - \sqrt{n})} = \frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} : \frac{m}{\sqrt{n}(\sqrt{m} - \sqrt{n})} = \frac{\sqrt{m}}{\sqrt{m} - \sqrt{n}} * \frac{\sqrt{n}(\sqrt{m} - \sqrt{n})}{m} = \frac{\sqrt{n}}{\sqrt{m}} = \sqrt{\frac{n}{m}}$

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

$\frac{a - 64}{\sqrt{a} + 3} * \frac{1}{a + 8\sqrt{a}} - \frac{\sqrt{a} + 8}{a - 3\sqrt{a}} = \frac{(\sqrt{a} - 8)(\sqrt{a} + 8)}{\sqrt{a} + 3} * \frac{1}{\sqrt{a}(\sqrt{a} + 8)} - \frac{\sqrt{a} + 8}{\sqrt{a}(\sqrt{a} - 3)} = \frac{\sqrt{a} - 8}{\sqrt{a} + 3} * \frac{1}{\sqrt{a}} - \frac{\sqrt{a} + 8}{\sqrt{a}(\sqrt{a} - 3)} = \frac{\sqrt{a} - 8}{\sqrt{a}(\sqrt{a} + 3)} - \frac{\sqrt{a} + 8}{\sqrt{a}(\sqrt{a} - 3)} = \frac{(\sqrt{a} - 8)(\sqrt{a} - 3) - (\sqrt{a} + 8)(\sqrt{a} + 3)}{\sqrt{a}(\sqrt{a} - 3)(\sqrt{a} + 3)} = \frac{a - 8\sqrt{a} - 3\sqrt{a} + 24 - (a + 8\sqrt{a} + 3\sqrt{a} + 24)}{\sqrt{a}(\sqrt{a} - 3)(\sqrt{a} + 3)} = \frac{a - 8\sqrt{a} - 3\sqrt{a} + 24 - a - 8\sqrt{a} - 3\sqrt{a} - 24}{\sqrt{a}(\sqrt{a} - 3)(\sqrt{a} + 3)} = \frac{-22\sqrt{a}}{\sqrt{a}(a - 9)} = -\frac{22}{a - 9}$