Question

Seja o complexo Z=2-6i encontre
A)o inverso de Z
B) o conjugado de Z elevado ao quadrado
C)o inverso de zxi

Answer 1

z = 2-6*i
a) Inverso de z: z^(-1) = 1/ (2-6i)
z^(-1) = (1 / 2-6i) * ((2+6i) / (2+6i))
z^(-1) = (2+6i) / ((2-6i)*(2+6i))
z^(-1) = (2+6i) / (4 + 12i - 12i - (6i)*(6i))
z^(-1) = (2+6i) / (4 - 36*i^2)
z^(-1) = (2+6i) / (4+36)
z^(-1) = (2+6i) / 40
z^(-1) = 2/40 + (6i)/40
z^(-1) = 1/20 + (3/20)*i

b) Conjugado de z^2:
Conjugado de z = 2+6i
(2+6i)^2 = (2+6i)*(2+6i) = 4+12i+12i-36 = -32 + 24i

c) Inverso de z vezes i
z*i = (2-6i)*i = 2i + 6. Logo
(z*i)^(-1) = 1/ (6+2i)
(z*i)^(-1) = (1/ (6+2i))*((6-2i) / (6-2i))
(z*i)^(-1) = (6-2i) / ((6+2i)*(6-2i))
(z*i)^(-1) = (6-2i) / (36 - 12i + 12i + 4)
(z*i)^(-1) = (6-2i) / (40)
(z*i)^(-1) = 6/40 - (2i)/40
(z*i)^(-1) = 3/20 - i / 20