$\text{I.}$

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

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

$\text{II.}$

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