Need help with Q on complex roots..

[VCE1996]

Hello

I need help with this question.. if someone could please show me how to do it and provide solutions that would be awesome

Find all the complex 5th roots of −i and sketch them in the complex plane. Answers must be left in exponential form.


Thanks

[Zealous]

Hello

I need help with this question.. if someone could please show me how to do it and provide solutions that would be awesome

Find all the complex 5th roots of −i and sketch them in the complex plane. Answers must be left in exponential form.


Thanks

Use De Moivre's Theorem:



Since we are taking the 5th root of -i, there will be 5 solutions so we can substitute k=0, 1, 2, 3, 4 in order to find these.











These are our five complex roots. You can then plot them on a complex plane and then put them into exponential form (re^itheta). I prefer polar form so I've left it like this.

[Thorium]

Hello

I need help with this question.. if someone could please show me how to do it and provide solutions that would be awesome

Find all the complex 5th roots of −i and sketch them in the complex plane. Answers must be left in exponential form.


Thanks

If the answer is to be given in exponential form, then I believe you have to use the calculator to solve the equation x^5=-i, as the power five is too high to be dealt with by hand. (and the answer will be a bunch of messy numbers :p) However, if we can express the ans in polar form, than it will be easier to do with hand.
Let r^5cis(5theta)=-i
r^5cis(5theta)=cis(-pi/2)
r=1
Theta=1/5(-pi/2+2kpi), k is Z
          =-pi/10, 3pi/10, 7pi/10, -9pi/10, -5pi/10

So fifth roots of -i in polar form will be cis(-pi/10), cis(3pi/10), cis(7pi/10), cis(-9pi/10), cis(-5pi/10)

I hope the graphing part will be easy. =)