$(\frac{8\sqrt{a}}{\sqrt{a} + 7} - \frac{15\sqrt{a}}{a + 14\sqrt{a} + 49}) : \frac{8\sqrt{a} + 41}{a - 49} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = (\frac{8\sqrt{a}}{\sqrt{a} + 7} - \frac{15\sqrt{a}}{(\sqrt{a} + 7)^2}) * \frac{a - 49}{8\sqrt{a} + 41} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{8\sqrt{a}(\sqrt{a} + 7) - 15\sqrt{a}}{(\sqrt{a} + 7)^2} * \frac{(\sqrt{a} - 7)(\sqrt{a} + 7)}{8\sqrt{a} + 41} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{8a + 56\sqrt{a} - 15\sqrt{a}}{\sqrt{a} + 7} * \frac{\sqrt{a} - 7}{8\sqrt{a} + 41} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{8a + 41\sqrt{a}}{\sqrt{a} + 7} * \frac{\sqrt{a} - 7}{8\sqrt{a} + 41} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{\sqrt{a}(8\sqrt{a} + 41)}{\sqrt{a} + 7} * \frac{\sqrt{a} - 7}{8\sqrt{a} + 41} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{\sqrt{a}}{\sqrt{a} + 7} * \frac{\sqrt{a} - 7}{1} + \frac{7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{a - 7\sqrt{a} + 7\sqrt{a} - 49}{\sqrt{a} + 7} = \frac{a - 49}{\sqrt{a} + 7} = \frac{(\sqrt{a} - 7)(\sqrt{a} + 7)}{\sqrt{a} + 7} = \sqrt{a} - 7$

$\frac{a\sqrt{a} + 27}{\sqrt{a} - \sqrt{b}} * (\frac{\sqrt{a} - 3}{a - 3\sqrt{a} + 9} - \frac{\sqrt{ab} - 9}{a\sqrt{a} + 27}) = \frac{(\sqrt{a})^3 + 3^3}{\sqrt{a} - \sqrt{b}} * (\frac{\sqrt{a} - 3}{a - 3\sqrt{a} + 9} - \frac{\sqrt{ab} - 9}{(\sqrt{a})^3 + 3^3}) = \frac{(\sqrt{a} + 3)(a - 3\sqrt{a} + 9)}{\sqrt{a} - \sqrt{b}} * (\frac{\sqrt{a} - 3}{a - 3\sqrt{a} + 9} - \frac{\sqrt{ab} - 9}{(\sqrt{a} + 3)(a - 3\sqrt{a} + 9)}) = \frac{(\sqrt{a} + 3)(a - 3\sqrt{a} + 9)}{\sqrt{a} - \sqrt{b}} * \frac{(\sqrt{a} - 3)(\sqrt{a} + 3) - (\sqrt{ab} - 9)}{(\sqrt{a} + 3)(a - 3\sqrt{a} + 9)} = \frac{1}{\sqrt{a} - \sqrt{b}} * \frac{a - 9 - \sqrt{ab} + 9}{1} = \frac{a - \sqrt{ab}}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a}(\sqrt{a} - \sqrt{b})}{\sqrt{a} - \sqrt{b}} = \sqrt{a}$