Question

Calcule o número complexo i^126 + i^126 + i^31 - i^180

Answer 1

$\mathtt{i^{126}+i^{126}+i^{31}-i^{180}}\\\\ =\mathtt{i^{124+2}+i^{124+2}+i^{28+3}-i^{180}}\\\\ =\mathtt{i^{4\cdot 31+2}+i^{4\cdot 31+2}+i^{4\cdot 7+3}-i^{4\cdot 45}}\\\\ =\mathtt{i^{4\cdot 31}\cdot i^2+i^{4\cdot 31}\cdot i^2+i^{4\cdot 7}\cdot i^3-i^{4\cdot 45}}\\\\ =\mathtt{(i^4)^{31}\cdot i^2+(i^4)^{31}\cdot i^2+(i^4)^7\cdot i^3-(i^4)^{45}}\\\\ =\mathtt{(1)^{31}\cdot (-1)+(1)^{31}\cdot (-1)+(1)^7\cdot (-i)-(1)^{45}}\\\\ =\mathtt{1\cdot (-1)+1\cdot (-1)+1\cdot (-i)-1}$

$=\mathtt{-1-1-i-1}$

$=\boxed{\begin{array}{c}\mathtt{-3-i} \end{array}}$   <———    esta é a resposta.


Dúvidas? Comente.


Bons estudos! :-)