Question

A expressão 1-i/1+i - 2i/1+3i, na qual i é a unidade imaginária é igual a:

Me ajudem por favor!

Answer 1

E aí Luan,

$\dfrac{1-i}{1+i}- \dfrac{2i}{1+3i}~~(multiplique~numerador~e~denominador~pelos~seus\\\\ conjugados)~\to~conjugado~de~1+i=1-i~~e~~de~1+3i=1-3i)\\\\\\ \dfrac{(1-i)(1-i)}{(1+i)(1-i)}- \dfrac{2i(1-3i)}{(1+3i)(1-3i)}~~\to~~ \dfrac{1-2i+i^2}{1-i^2}- \dfrac{2i-6i^2}{1-9i^2}\\\\\\ lembre-se~que~i^2=-1,~entao:\\\\\\ \dfrac{1-2i-1}{1-(-1)}- \dfrac{2i-6(-1)}{1-9(-1)}~~\to~~ \dfrac{-2i}{~~2}- \dfrac{6+2i}{10}~~MMC(2,10)=10$

$\dfrac{5(-2i)-1(6+2i)}{10}~~\to~~ \dfrac{-10i-6-2i}{10}~~\to~~ \dfrac{-6-12i}{10}\\\\\\ ~\to~ -\dfrac{6}{10}- \dfrac{12i}{10}~~\to~~ -\dfrac{3}{5}- \dfrac{6}{5}i$

Portanto,

$\large\boxed{\boxed{\dfrac{1-i}{1+i}- \dfrac{2i}{1+3i}= -\dfrac{3}{5}- \dfrac{6}{5}i}}\\ .$

Tenha ótimos estudos =))