$9^{-4} * 27^2 = (3^2)^{-4} * (3^3)^2 = 3^{-8} * 3^6 = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

$32^{-5} : 64^{-4} = (2^5)^{-5} : (2^6)^{-4} = 2^{-25} : 2^{-24} = 2^{-25 - (-24)} = 2^{-25 + 24} = 2^{-1} = \frac{1}{2}$

$(2\frac{7}{9})^{-7} * ((\frac{3}{5})^{-3})^{5} = (\frac{25}{9})^{-7} * (\frac{3}{5})^{-15} = (\frac{9}{25})^{7} * (\frac{5}{3})^{15} = (\frac{3^2}{5^2})^{7} * \frac{5^{15}}{3^{15}} = \frac{3^{14}}{5^{14}} * \frac{5^{15}}{3^{15}} = \frac{5}{3} = 1\frac{2}{3}$

$8^{-2} : 0,5^4 = (2^3)^{-2} : (2^{-1})^4 = 2^{-6} : 2^{-4} = 2^{-6 - (-4)} = 2^{-6 + 4} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

$\frac{22^6 * 2^{-8}}{44^{-3} * 11^9} = \frac{22^6 * 44^{3}}{2^{8} * 11^9} = \frac{(2 * 11)^6 * (2^2 * 11)^{3}}{2^{8} * 11^9} = \frac{2^6 * 11^6 * (2^2)^3 * 11^{3}}{2^{8} * 11^9} = \frac{2^6 * 2^6 * 11^{9}}{2^{8} * 11^9} = \frac{2^{12}}{2^{8}} = 2^4 = 16$

$\frac{10^{-2} * 15^{-4}}{30^{-6}} = \frac{30^{6}}{10^{2} * 15^{4}} = \frac{(2 * 3 * 5)^{6}}{(2 * 5)^{2} * (3 * 5)^{4}} = \frac{2^6 * 3^6 * 5^{6}}{2^2 * 5^{2} * 3^4 * 5^{4}} = \frac{2^4 * 3^2}{1} = 16 * 9 = 144$