$a^{-6}\ast a^9=a^{-6+9}=a^3$

$a^5\ast a^{-8}=a^{5-8}=a^{-3}$

$a^{-5}\ast a^{10}\ast a^{-12}=a^{-5+10-12}=a^{-7}$

$a^{-2}:a^6=a^{-2-6}=a^{-8}$

$a^7:a^{-3}=a^{7-(-3)}=a^{7+3}=a^{10}$

$a^{-3}:a^{-15}=a^{-3-(-15)}=a^{-3+15}=a^{12}$

$a^{12}\ast a^{-20}:a^{-9}=a^{12-20-(-9)}=a^{-8+9}=a^1$

$(a^{-5}{)}^4=a^{(-5)\ast 4}=a^{-20}$

$(a^{-6}{)}^{-8}=a^{-6\ast (-8)}=a^{48}$

$(a^2{)}^{-4}\ast (a^{-3}{)}^{-2}:(a^{-8}{)}^3=a^{2\ast (-4)+(-3)\ast (-2)-(-8)\ast 3}=a^{-8+6+24}=a^{22}$

$(a^4{b}^{-2}{c}^3{)}^{-10}=a^{4\ast (-10)}{b}^{-2\ast (-10)}{c}^{3\ast (-10)}=a^{-40}{b}^{20}{c}^{-30}$

$(\frac{a^{10}{b}^{-7}}{c^6{d}^{-14}}{)}^{-2}=\frac{a^{10\ast (-2)}{b}^{-7\ast (-2)}}{c^{6\ast (-2)}{d}^{-14\ast (-2)}}=\frac{a^{-20}{b}^{14}}{c^{-12}{d}^{28}}=a^{-20}{b}^{14}{c}^{12}{d}^{-28}$