$\frac{4a + 4\sqrt{5}}{a^2 - 5} = \frac{4(a + \sqrt{5})}{(a - \sqrt{5})(a + \sqrt{5})} = \frac{4}{a -\sqrt{5}}$

$\frac{\sqrt{28} - 2\sqrt{2a}}{6a - 21} = \frac{\sqrt{4 * 7} - 2\sqrt{2a}}{3(2a - 7)} = \frac{2\sqrt{7} - 2\sqrt{2a}}{3((\sqrt{2a})^2 - (\sqrt{7})^2)} = \frac{2(\sqrt{7} - \sqrt{2a})}{3(\sqrt{2a} - \sqrt{7})(\sqrt{2a} + \sqrt{7})} = -\frac{2(\sqrt{2a} - \sqrt{7})}{3(\sqrt{2a} - \sqrt{7})(\sqrt{2a} + \sqrt{7})} = -\frac{2}{3(\sqrt{2a} + \sqrt{7})}$

$\frac{a + 4\sqrt{ab} + 4b}{a - 4b} = \frac{(\sqrt{a})^2 + 2 * \sqrt{a} * 2\sqrt{b} + (2\sqrt{b})^2}{(\sqrt{a})^2 - (2\sqrt{b})^2} = \frac{(\sqrt{a} + 2\sqrt{b})^2}{(\sqrt{a} - 2\sqrt{b})(\sqrt{a} + 2\sqrt{b})} = \frac{\sqrt{a} + 2\sqrt{b}}{\sqrt{a} - 2\sqrt{b}}$, если a > 0, b > 0

$\frac{x^2 - 6y}{x^2 + 6y - x\sqrt{24y}} = \frac{x^2 - (\sqrt{6y})^2}{x^2 - x\sqrt{4 * 6y} + (\sqrt{6y})^2} = \frac{(x - \sqrt{6y})(x + \sqrt{6y})}{x^2 - 2x\sqrt{6y} + (\sqrt{6y})^2} = \frac{(x - \sqrt{6y})(x + \sqrt{6y})}{(x - \sqrt{6y})^2} = \frac{x + \sqrt{6y}}{x - \sqrt{6y}}$

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

$\frac{m\sqrt{m} - 27}{\sqrt{m} - 3} = \frac{\sqrt{m^2 * m} - 27}{\sqrt{m} - 3} = \frac{\sqrt{m^3} - 27}{\sqrt{m} - 3} = \frac{(\sqrt{m})^3 - 3^3}{\sqrt{m} - 3} = \frac{(\sqrt{m} - 3)((\sqrt{m})^2) + 3\sqrt{m} + 3^2)}{\sqrt{m} - 3} = m + 3\sqrt{m} + 9$