Question

Упростите выражение:
Комплексные числа
$i {}^{41}$
$i {}^{33} + i {}^{7}$
​

Answer 1

$\displaystyle\bf\\i^{1} =i \ \ ; \ \ i^{2}=-1 \ \ ; \ \ i^{3} =i\cdot i^{2} =-i \ \ ; \ \ i^{4} =i^{3} \cdot i=-i\cdot i=1\\\\\\i^{4k} =1 \ \ ; \ \ i^{4k+1} =i \ \ ; \ \ i^{4k+2}=-1 \ \ ; \ \ i^{4k+3} =-i\\\\\\i^{41} =(i^{4})^{10} \cdot i=1^{10} \cdot i=i\\\\\\i^{33} =(i^{4} )^{8}\cdot i=1^{8} \cdot i=i\\\\\\i^{7} =i^{3} \cdot i^{4} =-i\cdot 1=-i\\\\\\i^{33} +i^{7} =i+(-i)=0$