$a)\frac{7^{-4}\cdot 7^5}{7^2}=\frac{7^{-4+5}}{7^2}=\frac{7^1}{7^2}=7^{1-2}=7^{-1}=\frac{1}{7}$  

$b)\frac{{\left(-\frac{4}{5}\right)}^{-3}\cdot {\left(-\frac{5}{4}\right)}^{-1}}{{\left(-\frac{4}{5}\right)}^{-5}}=\frac{{\left(-\frac{4}{5}\right)}^{-3}\cdot {\left(-\frac{4}{5}\right)}^1}{{\left(-\frac{4}{5}\right)}^{-5}}=\frac{{\left(-\frac{4}{5}\right)}^{-2}}{{\left(-\frac{4}{5}\right)}^{-5}}={\left(-\frac{4}{5}\right)}^{-2-\left(-5\right)}={\left(-\frac{4}{5}\right)}^{-2+5}={\left(-\frac{4}{5}\right)}^3=-\frac{64}{125}$  

$c)\frac{{\left({\left(-4\frac{1}{5}\right)}^2\right)}^{-4}\cdot {\left({\left(-4,2\right)}^{-3}\right)}^2}{{\left(-\frac{21}{5}\right)}^{-12}}=\frac{{\left(-4\frac{1}{5}\right)}^{2\cdot \left(-4\right)}\cdot {\left(-4\frac{1}{5}\right)}^{-3\cdot 2}}{{\left(-4\frac{1}{5}\right)}^{-12}}=\frac{{\left(-4\frac{1}{5}\right)}^{-8-6}}{{\left(-4\frac{1}{5}\right)}^{-12}}={\left(-4\frac{1}{5}\right)}^{-14-\left(-12\right)}={\left(-4\frac{1}{5}\right)}^{-2}={\left(-\frac{21}{5}\right)}^{-2}={\left(\frac{5}{21}\right)}^2=\frac{25}{441}$  

 

$\text{d)}{\left(\frac{{\left({\left(2,56\right)}^{\frac{1}{2}}\cdot {\left(2,56\right)}^{-\frac{3}{2}}\right)}^2}{{\left(2,56\right)}^{-\frac{1}{6}}\cdot {\left(2,56\right)}^{-\frac{5}{6}}}\right)}^{-\frac{1}{3}}={\left(\frac{{\left({\left(2,56\right)}^{\frac{1}{2}+\left(-\frac{3}{2}\right)}\right)}^2}{{\left(2,56\right)}^{-\frac{1}{6}+\left(-\frac{5}{6}\right)}}\right)}^{-\frac{1}{3}}={\left(\frac{{\left(2,56\right)}^{-1\cdot 2}}{{\left(2,56\right)}^{-1}}\right)}^{-\frac{1}{3}}=$  

$={\left(\frac{{\left(2,56\right)}^{-2}}{{\left(2,56\right)}^{-1}}\right)}^{-\frac{1}{3}}={\left({\left(\frac{256}{100}\right)}^{-2-\left(-1\right)}\right)}^{-\frac{1}{3}}={\left({\left(\frac{256}{100}\right)}^{-1}\right)}^{-\frac{1}{3}}={\left(\frac{64}{25}\right)}^{-1\cdot \left(-\frac{1}{3}\right)}={\left(\frac{64}{25}\right)}^{\frac{1}{3}}$