$\text{I.}$

${\left(-0,0016\right)}^4:{0,00008}^4={\left(-0,0016:0,00008\right)}^4={\left(-\frac{16}{10000}:\frac{8}{100000}\right)}^4={\left(-\frac{{\sout{16}}^2}{{\sout{10000}}^1}\cdot \frac{{\sout{100000}}^{10}}{{\sout{8}}^1}\right)}^4=$

$={\left(-20\right)}^4={20}^4$

 

$\text{II.}$

$\frac{5^3}{6^3}:\frac{{25}^3}{{42}^3}={\left(\frac{5}{6}:\frac{25}{42}\right)}^3={\left(\frac{{\sout{5}}^1}{{\sout{6}}^1}\cdot \frac{{\sout{42}}^7}{{\sout{25}}^5}\right)}^3=$ ${\left(\frac{7}{5}\right)}^3$

 

$\text{III.}$

$\frac{{\left(-0,48\right)}^4:{0,12}^4}{{1,2}^4:{\left(-0,6\right)}^4}=$ $\frac{{\left(-0,48:0,12\right)}^4}{{\left(1,2:\left(-0,6\right)\right)}^4}=$ ${\left(\frac{-0,48:0,12}{1,2:\left(-0,6\right)}\right)}^4={\left(\frac{-4}{-2}\right)}^4=2^4$

 

$\text{IV.}$

$\frac{{\left(\frac{1}{2}\right)}^{100}:{1,5}^{100}:{\left(\frac{5}{6}\right)}^{100}}{{\left(\frac{5}{6}\right)}^{100}:{\left(2\frac{1}{12}\right)}^{100}}=$ ${\left(\frac{\frac{1}{2}:\frac{3}{2}:\frac{5}{6}}{\frac{5}{6}:2\frac{1}{12}}\right)}^{100}$ $={\left(\frac{\frac{1}{{\sout{2}}^1}\cdot \frac{{\sout{2}}^1}{{\sout{3}}^1}\cdot \frac{{\sout{6}}^2}{5}}{\frac{5}{6}:\frac{25}{12}}\right)}^{100}=$ ${\left(\frac{\frac{2}{5}}{\frac{{\sout{5}}^1}{{\sout{6}}^1}\cdot \frac{{\sout{12}}^2}{\sout{25}{}^5}}\right)}^{100}=$

$={\left(\frac{\frac{2}{5}}{\frac{2}{5}}\right)}^{100}={\left(\frac{{\sout{2}}^1}{{\sout{5}}^1}\cdot \frac{{\sout{5}}^1}{{\sout{2}}^1}\right)}^{100}=1^{100}=1$  

 

$\text{V.}$

$\frac{{\left(\frac{6}{7}\right)}^8\cdot {\left(\frac{14}{27}\right)}^8}{{\left(\frac{2}{9}\right)}^8}={\left(\frac{\frac{6}{{\sout{7}}^1}\cdot \frac{{\sout{14}}^2}{27}}{\frac{2}{9}}\right)}^8=$ ${\left(\frac{\frac{12}{27}}{\frac{2}{9}}\right)}^8={\left(\frac{{\sout{12}}^6}{{\sout{27}}^3}\cdot \frac{{\sout{9}}^1}{{\sout{2}}^1}\right)}^8=2^8$