Question

Find all solutions for the complex numbers:

a) z^4 = 9i -46
b) 8*sqrt3 / z^4 +8 =+i

Answer 1

best convert to polar form

z^4 = 46.87 cis 2.95
so, z = 2.62 cis 0.738 + k*pi/2 for k=0..3
then convert back to rectangular form.

Answer 2

using de Moivre's Theorem:

let z^4 = -46 + 9i

changing -46 + 9i to rectangular form
r = √46^2+9^2 =√2197
tanØ = 9/-46
Ø = 168.9298°

z^4 = √2197(cos168.9298° + i sin 168.9298°)
z = .....
so there will be 4 roots,

z = √2197^(1/4) (cos (168.9.. +k(360°)/4 + i sin(168.9..+k(360°)/4 )
where k = 0, 1, 2, 3

if k = 0

z = √2197^(1/4) (cos42.2324 + i sin42.2324)
= 2.61655(.740424 , .67214 i )
= (1.937355 , 1.758687 i )

if k = 1
z = √2197(1/4) (cos 132.2324° + i sin 132.2324°)
= ( -1.758687 , 1.937355 i)

k = 2
I will let you finish the other two.

testing for k = 1

if z = ( -1.758687 , 1.937355 i)
r = appr 2.61655
Ø = 132.2324
then z^4 = 2.6155^4 (cos 4(132.2324) , i sin 4(132.2324))
= 46.872...(-.98139.. , .19201..)
= -46 + 9i , YEAAAAHHH