$6^{-9}\ast 6^6=6^{-9+6}=6^{-3}=\frac{1}{6^3}=\frac{1}{216}$

$7^{-16}:7^{-18}=7^{-16-(-18)}=7^{-16+18}=7^2=49$

$5^{-7}:5^{-6}\ast 5^3=5^{-7-(-6)+3}=5^{-7+6+3}=5^2=25$

$\frac{4^{-7}\ast (4^{-5}{)}^3}{(4^{-3}{)}^7}=\frac{4^{-7-5\ast 3}}{4^{-3\ast 7}}=\frac{4^{-7-15}}{4^{-21}}=\frac{4^{-22}}{4^{-21}}=4^{-22-(-21)}=4^{-22+21}=4^{-1}=\frac{1}{4}$

$0,8^{-4}\ast (1\frac{1}{4}{)}^{-4}=(\frac{8}{10}{)}^{-4}\ast (\frac{5}{4}{)}^{-4}=(\frac{4}{5}{)}^{-4}\ast (\frac{5}{4}{)}^{-4}=\frac{4^{-4}}{5^{-4}}\ast \frac{5^{-4}}{4^{-4}}=1$

$\frac{{11}^{-2}}{{22}^{-2}}=\frac{{11}^{-2}}{(2\ast 11{)}^{-2}}=\frac{{11}^{-2}}{2^{-2}\ast {11}^{-2}}=\frac{1}{2^{-2}}=2^2=4$