let z1= 2 − 3i and z2 = −1 + 2i.
compute:
(z1+z2)^3
and
(z1-conguate of z2)^-2

Question

let z1= 2 − 3i and z2 = −1 + 2i.
compute:
(z1+z2)^3
and
(z1-conguate of z2)^-2

experimentx · Answer

z1+z2 =  2 − 3i −1 + 2i = 2 −1− 3i + 2i = 1 - i

now we change 1 - i into a form where DeMoivre's theorem can be applied
$$1 - i =\sqrt{2}(1/\sqrt{2} - i/\sqrt{2})$$
            = $$\sqrt{2}(\cos(-\pi/4) + i \sin (-\pi/4))$$

(1-i)^3 = $$2^{3/2}(\cos(- 3 \pi/4) + \sin(-3\pi/4))$$

zarkon · Answer

simplifies nicely to -2-2i

experimentx · Answer

(z1-conguate of z2)^-2 = (2 − 3i −1 - 2i)^-2 = 1/(1-5i)^2
(1-5i)^2 simplify this .. ==> (1-25)+(-10)i => -24 - 10 i

=-1/(24+10i)
multiply both numerator and denominator by conjugate (24 - 10i) ... that should simplify the answer.