$\frac{x - 25}{\sqrt{x} - 5} = \frac{(\sqrt{x})^2 - 5^2}{\sqrt{x} - 5} = \frac{(\sqrt{x} - 5)(\sqrt{x} + 5)}{\sqrt{x} - 5} = \sqrt{x} + 5$

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

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

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

$\frac{23 - \sqrt{23}}{\sqrt{23}} = \frac{(\sqrt{23})^2 - \sqrt{23}}{\sqrt{23}} = \frac{\sqrt{23}(\sqrt{23} - 1)}{\sqrt{23}} = \sqrt{23} - 1$

$\frac{\sqrt{24} - \sqrt{28}}{\sqrt{54} - \sqrt{63}} = \frac{\sqrt{4 * 6} - \sqrt{4 * 7}}{\sqrt{9 * 6} - \sqrt{9 * 7}} = \frac{2\sqrt{6} - 2\sqrt{7}}{3\sqrt{6} - 3\sqrt{7}} = \frac{2(\sqrt{6} - \sqrt{7})}{3(\sqrt{6} - \sqrt{7})} = \frac{2}{3}$

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

$\frac{b - 8\sqrt{b} + 16}{\sqrt{b} - 4} = \frac{(\sqrt{b})^2 - 2 * \sqrt{b} * 4 + 4^2}{\sqrt{b} - 4} = \frac{(\sqrt{b} - 4)^2}{\sqrt{b} - 4} = \sqrt{b} - 4$