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

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

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

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

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

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