SOLUTION: Hi, i have a problem which states Find all numbers z in C such that ((z-i)^4)=i Help would be much appreciated, thank you so much!

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson  -> Lesson -> SOLUTION: Hi, i have a problem which states Find all numbers z in C such that ((z-i)^4)=i Help would be much appreciated, thank you so much!      Log On


   

Question 1074556: Hi, i have a problem which states Find all numbers z in C such that ((z-i)^4)=i
Help would be much appreciated, thank you so much!
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
%28z-i%29%5E4=i

First find the 4 fourth roots of i, which are

1%28cos%28pi%2F2%2B2pi%2An%29%5E%22%22%2Bi%2Asin%28pi%2F2%2B2pi%2An%29%29, for n=0,1,2,3

cos%28%28%281%2B4n%29%2F2%29%5E%22%22pi%29%2Bi%2Asin%28%28%281%2B4n%29%2F2%29%5E%22%22pi%29. for n=0,1,2,3

Then taking 4th roots of both sides we have:

, for n=0,1,2,3

So

, for n=0,1,2,3 

Now substitute n=0,1,2, and 3 and simplify and you'll have the 
4 solutions.

Edwin