Question

Compute the absolute value and the conjugate of z = (1 + i) ^ 6​

Answer 1

z = (1 + i)6 =³√2(cos π4+ isinπ4)´6= 8 µcos3π2+ isin3π2)¶= −8i. Hence |z| = 8 and ¯z = 8i. z = (1 + i)

I think I am not real sure

Answer 2

In polar form, we have

`1+i = \sqrt2 e^(i\pi/4)`

Then by DeMoivre's theorem,

`(1+i)^6 = \left(\sqrt2\right)^6 e^(i\,6\pi/4) = 2^3 e^(i\,3\pi/2) = -8i`

Then

`|z| = |-8i| = \boxed{8}`

and

`\bar z = \overline{-8i} = \boxed{8i}`