$a^6\ast a^{-10}=a^{6-10}=a^{-4}$

$a^4:a^7=a^{4-7}=a^{-3}$

$a^{-5}:a^{-9}=a^{-5-(-9)}=a^{-5+9}=a^4$

$(a^{-2}{)}^6=a^{-2\ast 6}=a^{-12}$

$(a^{-3}{b}^{-1}{c}^7{)}^{-4}=a^{-3\ast (-4)}{b}^{-1\ast (-4)}{c}^{7\ast (-4)}=a^{12}{b}^4{c}^{-28}$

$(\frac{a^2}{b{c}^{-1}}{)}^{-3}=\frac{a^{2\ast (-3)}}{b^{-3}{c}^{-1\ast (-3)}}=\frac{a^{-6}}{b^{-3}{c}^3}=a^{-6}{b}^3{c}^{-3}$

$a^{-16}\ast a^8:a^{-4}=a^{-16+8-(-4)}=a^{-8+4}=a^{-4}$

$(a^{-3}{)}^8:(a^{-1}{)}^7\ast (a^{-7}{)}^{-4}=a^{-3\ast 8-(-1)\ast 7+(-7)\ast (-4)}=a^{-24+7+28}=a^{11}$