$\frac{x^2 - 19}{x + \sqrt{19}} = \frac{x^2 - (\sqrt{19})^2}{x + \sqrt{19}} = \frac{(x - \sqrt{19})(x + \sqrt{19})}{x + \sqrt{19}} = x - \sqrt{19}$

$\frac{\sqrt{x} - 6}{x - 36} = \frac{\sqrt{x} - 6}{(\sqrt{x})^2 - 6^2} = \frac{\sqrt{x} - 6}{(\sqrt{x} - 6)(\sqrt{x} + 6)} = \frac{1}{\sqrt{x} + 6}$

$\frac{m + 8\sqrt{m}}{m - 64} = \frac{(\sqrt{m})^2 + 8\sqrt{m}}{(\sqrt{m})^2 - 8^2} = \frac{\sqrt{m}(\sqrt{m} + 8)}{(\sqrt{m} - 8)(\sqrt{m} + 8)} = \frac{\sqrt{m}}{\sqrt{m} - 8}$

$\frac{29 - \sqrt{29}}{\sqrt{29}} = \frac{(\sqrt{29})^2 - \sqrt{29}}{\sqrt{29}} = \frac{\sqrt{29}(\sqrt{29} - 1)}{\sqrt{29}} = \sqrt{29} - 1$

$\frac{a - 6\sqrt{ab} + 9b}{a - 9b} = \frac{(\sqrt{a})^2 - 2 * 3\sqrt{ab} + (3\sqrt{b})^2}{(\sqrt{a})^2 - (3\sqrt{b})^2} = \frac{(\sqrt{a} - 3\sqrt{b})^2}{(\sqrt{a} - 3\sqrt{b})(\sqrt{a} + 3\sqrt{b})} = \frac{\sqrt{a} - 3\sqrt{b}}{\sqrt{a} + 3\sqrt{b}}$, если a > 0, b > 0

$\frac{11 - \sqrt{33}}{\sqrt{33} - 3} = \frac{(\sqrt{11})^2 - \sqrt{11} * \sqrt{3}}{\sqrt{11} * \sqrt{3} - (\sqrt{3})^2} = \frac{\sqrt{11}(\sqrt{11} - \sqrt{3})}{\sqrt{3}(\sqrt{11} - \sqrt{3})} = \frac{\sqrt{11}}{\sqrt{3}} = \sqrt{\frac{11}{3}} = \sqrt{3\frac{2}{3}}$