$$0,5^{-1}+0,{25}^{-2}+0,{125}^{-3}+0,{0625}^{-4}$$

$$={\left(\frac{5}{10}\right)}^{-1}+{\left(\frac{25}{100}\right)}^{-2}+{\left(\frac{125}{1000}\right)}^{-3}+{\left(\frac{625}{10000}\right)}^{-4}$$

$$={\left(\frac{1}{2}\right)}^{-1}+{\left(\frac{1}{4}\right)}^{-2}+{\left(\frac{1}{8}\right)}^{-3}+{\left(\frac{1}{16}\right)}^{-4}$$

$$={\left(\frac{2}{1}\right)}^1+{\left(\frac{4}{1}\right)}^2+{\left(\frac{8}{1}\right)}^3+{\left(\frac{16}{1}\right)}^4$$

$$=\frac{2}{1}+\frac{16}{1}+\frac{512}{1}+\frac{65536}{1}$$

$$=2+16+512+65536$$