Question

Considerando a unidade imaginaria i , pose afirmar que o Número complexo (1-i/1+i)⁵ e igual a

Answer 1

Simplificar a expressão [(1-i)/(1+i)]^5:

$\left(\frac{1-i}{1+i} \right)^5\\\\= \left(\frac{1-i}{1+i} \right)^4 \times \frac{1-i}{1+i}\\\\= \left[ \left(\frac{1-i}{1+i} \right)^2 \right]^2 \times \frac{1-i}{1+i}\\\\= \left( \frac{1 - 2i + i^2}{1 +2i +i^2} \right)^2 \times \frac{1-i}{1+i}\\\\= \left( \frac{1 - 2i -1}{1 +2i -1} \right)^2 \times \frac{1-i}{1+i}\\\\= \left( \frac{- 2i}{2i } \right)^2 \times \frac{1-i}{1+i}\\\\= \left( -1 \right)^2 \times \frac{1-i}{1+i}\\\\= 1 \times \frac{1-i}{1+i}\\\\= \frac{1-i}{1+i}$

$= \frac{1-i}{1+i} \times \frac{1-i}{1-i}\\\\= \frac{\left( 1 - i \right)^2}{1 - i^2}\\\\= \frac{1 - 2i + i^2}{1 - (-1)}\\\\= \frac{1 - 2i -1}{1 + 1}\\\\= \frac{-2i}{2}\\\\= -i.$