Question

Can someone help me with: z^3 = 27i ?

Answer 1

What is the symbol after Z supposed to be?

Answer 2

you need to find z value, don't you?

Answer 3

elevated in third Pepitop

Answer 4

Yes NooriK

Answer 5

do you know De Moivre's theorem?

Answer 6

yes I know, but I can't figure out what to do..

Answer 7

first you have to find mod (z)
mod(z)=sqrt(x^2+y^2)
mod(z)=sqrt(0+27^2)
then
\[\cos \theta = \frac{ x }{ \mod(z) }\]
\[\sin \theta = \frac{ x }{ \mod(z) }\]
then \[\arg(z) = \theta\]
then \[z^3 = mod(z) * (\cos\theta +\sin\theta)\]
then \[z =\sqrt[3]{\mod(z)}*(\cos \frac{ \theta+2k \pi }{ 3 }+i \sin \frac{ \theta+2k \pi }{ 3 })\]
it's hard, i know that, but i hope that'll help

Answer 8

Thanks! Very helpful

Answer 9

you must note that, in the last form its cos ((mod(z)+2piK)/3)+i sin((mod(z)+2piK)/3)

Answer 10

you are welcome

Answer 11

What is θ here?

Answer 12

you know that we have transformed it into the polar form, so theta is an angle

Answer 13

yeah I know but how do you find it, i'm very newbie at this kind of math..