Question

Calcule (1+i)^10 – (1- i)^10

Answer 1

Resposta:

(1+i)¹⁰-(1-i)¹⁰=64i

Explicação passo-a-passo:

(1+i)¹⁰-(1-i)¹⁰=[(1+i)²]⁵-[(1-i)²]⁵

(1+i)²=1²+2i+i²=1+2i-1=2i

(1-i)²=1²-2i+i²=1-2i-1= -2i

[(1+i)²]⁵-[(1-i)²]⁵=[2i]⁵-[-2i]⁵=2⁵.i⁵-[(-1)⁵.2⁵.i⁵]

i⁵=i⁴.i¹=i

(-1)⁵= -1

2⁵.i⁵-[(-1)⁵.2⁵.i⁵]=2⁵.i-[-2⁵.i]=2⁵.i+2⁵.i=2.2⁵.i=2⁶i=64i