Question

Calcule o conjugado do número complexo Z, tal que Z + 2i^81 = -3Z + i^26

Answer 1

$Lembrando:~~\left\{\begin{array}{ccc}i^1&=&i\\i^2&=&-1\\i^3&=&-i\\i^4&=&1\end{array}\right$

$Z~+~2i^{81}~=~-3Z~+~i^{26}\\\\\\Z+3Z~=~i^{26}~-~2i^{81}\\\\\\4Z~=~i^{26}~-~2i^{81}\\\\\\\boxed{Z~=~\frac{i^{26}~-~2i^{81}}{4}}\\\\\\\\\\\rightarrow~~i^{26}~=~i^{6~.~4~+~2}~=~\left(i^4\right)^6~.~i^2~=~(1)^6~.~(-1)~=~\boxed{-1}\\\\\\\rightarrow~~i^{81}~=~i^{20~.~4~+~1}~=~\left(i^4\right)^{20}~.~i^1~=~(1)^{20}~.~(i)~=~\boxed{i}\\\\\\\\Z~=~\frac{(-1)~-~2\,.\,i}{4}\\\\\\Z~=~\frac{-1~-~2i}{4}\\\\\\\boxed{Z~=~-\frac{1}{4}~-~i.\frac{1}{2}}$

Podemos agora determinar o conjugado de Z:

$\overline{Z}~=~\overline{\left(-\frac{1}{4}~-~i.\frac{1}{2}\right)}\\\\\\\boxed{\overline{Z}~=~\left(-\frac{1}{4}~+~i.\frac{1}{2}\right)}$