Question

Sendo i a unidade imaginária, (1+i)^-2 é igual a:
a)1
b)-i
c)2.i
d)-i/2
e)i/2

Answer 1

Usaremos as seguintes propriedades:

$\boxed{\boxed{i^{2}=-1}}\\\\\\\boxed{\boxed{a^{-b}=\dfrac{1}{a^{b}}~~~para~a\neq0}}\\\\\\\boxed{\boxed{(a+b)^{2}=a^{2}+2ab+b^{2}}}$
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$(1+i)^{-2}=\dfrac{1}{(1+i)^{2}}$

Expandindo $(1+i)^{2}$:

$(1+i)^{-2}=\dfrac{1}{1^{2}+2\cdot1\cdot i+i^{2}}\\\\\\(1+i)^{-2}=\dfrac{1}{1+2i+i^{2}}\\\\\\(1+i)^{-2}=\dfrac{1}{1+2i-1}\\\\\\(1+i)^{-2}=\dfrac{1}{2i}$

Multiplicando o numerador e o denominador por i:

$(1+i)^{-2}=\dfrac{i}{2i^{2}}\\\\\\(1+i)^{-2}=\dfrac{i}{2\cdot(-1)}\\\\\\\boxed{\boxed{(1+i)^{-2}=-\dfrac{i}{2}}}$