Question

Sa se rezolve urmatorul calcul cu numere complexe: i^240+i^2021
Dau coroana!

Answer 1

 

$\displaystyle\bf\\Aplicam~proprietatile~puterilor.\\\\i^{240}+i^{2021}=\\\\=i^{240}+i^{2020+1}=\\\\=i^{2\times2\times60}+i^{2\times2\times505}\times i^1=\\\\=\Big(i^2\Big)^{2\times60}+\Big(i^2\Big)^{2\times505}\times i^1=\\\\=\Big(-1\Big)^{2\times60}+\Big(-1\Big)^{2\times505}\times i=\\\\=\Big((-1)^2\Big)^{60}+\Big((-1)^2\Big)^{505}\times i=\\\\=\Big(1\Big)^{60}+\Big(1\Big)^{505}\times i=\\\\=1+1\times i=\\\\=\boxed{\bf1+i}$