$a){\left(7^{-2}\right)}^3\cdot {\left(7^4\right)}^{-1}=7^{-2\cdot 3}\cdot 7^{4\cdot \left(-1\right)}=7^{-6}\cdot 7^{-4}=7^{-6+\left(-4\right)}=7^{-10}$  

$b){\left({\left(6^{-3}\right)}^2\right)}^{-4}=6^{-3\cdot 2\cdot \left(-4\right)}=6^{24}$ $b){\left({\left(6^{-3}\right)}^2\right)}^{-4}=6^{-3\cdot 2\cdot \left(-4\right)}=6^{24}$  

$c)\frac{{\left(5^3\right)}^{-1}\cdot {\left(5^{-3}\right)}^{-2}}{5^2}=$   $\frac{5^{3\cdot \left(-1\right)}\cdot 5^{-3\cdot \left(-2\right)}}{5^2}=\frac{5^{-3}\cdot 5^6}{5^2}=$   $\frac{5^{-3+6}}{5^2}=\frac{5^3}{5^2}=5^1$  

$d)\frac{{\left(9^3\right)}^{-2}:9^{-4}}{9\cdot {\left(9^{-2}\right)}^{-1}}=$    $\frac{9^{3\cdot \left(-2\right)}:9^{-4}}{9^1\cdot 9^{-2\cdot \left(-1\right)}}=\frac{9^{-6}:9^{-4}}{9^1\cdot 9^2}=\frac{9^{-6-\left(-4\right)}}{9^{1+2}}=\frac{9^{-6+4}}{9^3}=\frac{9^{-2}}{9^3}=9^{-2-3}=9^{-5}$