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

$\frac{\sqrt{m} + 1}{\sqrt{m} - 2} - \frac{\sqrt{m} + 3}{\sqrt{m}} = \frac{\sqrt{m}(\sqrt{m} + 1) - (\sqrt{m} + 3)(\sqrt{m} - 2)}{\sqrt{m}(\sqrt{m} - 2)} = \frac{m + \sqrt{m} - (m + 3\sqrt{m} - 2\sqrt{m} - 6)}{\sqrt{m}(\sqrt{m} - 2)} = \frac{m + \sqrt{m} - m - 3\sqrt{m} + 2\sqrt{m} + 6}{\sqrt{m}(\sqrt{m} - 2)} = \frac{6}{m - 2\sqrt{m}}$

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

$\frac{\sqrt{a}}{\sqrt{a} + 4} - \frac{a}{a - 16} = \frac{\sqrt{a}}{\sqrt{a} + 4} - \frac{a}{(\sqrt{a})^2 - 4^2} = \frac{\sqrt{a}}{\sqrt{a} + 4} - \frac{a}{(\sqrt{a} - 4)(\sqrt{a} + 4)} = \frac{\sqrt{a}(\sqrt{a} - 4) - a}{(\sqrt{a} - 4)(\sqrt{a} + 4)} = \frac{a - 4\sqrt{a} - a}{a - 16} = \frac{-4\sqrt{a}}{a - 16} = -\frac{4\sqrt{a}}{a - 16}$

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

$\frac{a + \sqrt{a}}{\sqrt{b}} * \frac{b}{2\sqrt{a} + 2} = \frac{(\sqrt{a})^2 + \sqrt{a}}{\sqrt{b}} * \frac{(\sqrt{b})^2}{2(\sqrt{a} + 1)} = \frac{\sqrt{a}(\sqrt{a} + 1)}{1} * \frac{\sqrt{b}}{2(\sqrt{a} + 1)} = \frac{\sqrt{a}}{1} * \frac{\sqrt{b}}{2} = \frac{\sqrt{ab}}{2}$

$\frac{\sqrt{c} - 5}{\sqrt{c}} : \frac{с - 25}{3с} = \frac{\sqrt{c} - 5}{\sqrt{c}} * \frac{3с}{с - 25} = \frac{\sqrt{c} - 5}{\sqrt{c}} * \frac{3 * (\sqrt{c})^2}{(\sqrt{c})^2 - 5^2} = \frac{\sqrt{c} - 5}{1} * \frac{3\sqrt{c}}{(\sqrt{c} - 5)(\sqrt{c} + 5)} = \frac{3\sqrt{c}}{\sqrt{c} + 5}$

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

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

$(\frac{\sqrt{x} - 3}{\sqrt{x} + 3} + \frac{12\sqrt{x}}{x - 9}) : \frac{\sqrt{x} + 3}{x - 3\sqrt{x}} = (\frac{\sqrt{x} - 3}{\sqrt{x} + 3} + \frac{12\sqrt{x}}{(\sqrt{x})^2 - 3^2}) * \frac{x - 3\sqrt{x}}{\sqrt{x} + 3} = \frac{(\sqrt{x} - 3)^2 + 12\sqrt{x}}{(\sqrt{x} - 3)(\sqrt{x} + 3)} * \frac{(\sqrt{x})^2 - 3\sqrt{x}}{\sqrt{x} + 3} = \frac{(\sqrt{x})^2 - 2 * 3\sqrt{x} + 3^2 + 12\sqrt{x}}{(\sqrt{x} - 3)(\sqrt{x} + 3)} * \frac{\sqrt{x}(\sqrt{x} - 3)}{\sqrt{x} + 3} = \frac{x - 6\sqrt{x} + 9 + 12\sqrt{x}}{\sqrt{x} + 3} * \frac{\sqrt{x}}{\sqrt{x} + 3} = \frac{x + 6\sqrt{x} + 9}{\sqrt{x} + 3} * \frac{\sqrt{x}}{\sqrt{x} + 3} = \frac{(\sqrt{x})^2 + 2 * 3\sqrt{x} + 3^2}{\sqrt{x} + 3} * \frac{\sqrt{x}}{\sqrt{x} + 3} = \frac{(\sqrt{x} + 3)^2}{\sqrt{x} + 3} * \frac{\sqrt{x}}{\sqrt{x} + 3} = \sqrt{x}$