Cube roots of a complex number

[fortwrestler]

I am having trouble right now. I have a quiz tomorrow in pre-cal and I am kinda lost after the winter break and I am stuck on one of my homework problems.

It states:

"Find the cube roots of (sorry i dont know how to do this) negative square root of three plus i(the i is not included in the square root) in polar form"

Please help me if you can

 

[fortwrestler]

maybe this looks better

"find the cube roots of -3^(1/2) +i in polar form"

 

[soroban]

Hello, fortwrestler!

Exactly where is your difficulty?

Find the cube roots of \(\displaystyle z\:=\:-\sqrt{3}\,+\,i\) in polar form.

Having trouble writing \(\displaystyle z\) in polar form?

    *       |
    : \     |
   1:   \2  | 
    :     \ |
  - + - - - + - - -
      -√3

We have: \(\displaystyle \,r\,=\,2\) and \(\displaystyle \theta\,=\,\frac{5\pi}{6}\)

\(\displaystyle \;\;\)So: \(\displaystyle \L\,z\;=\;2\left(\cos\frac{5\pi}{6}\,+\,i\cdot\sin\frac{5\pi}{6}\right)\)


Now can you apply DeMoivre's (Euler's?) Formula to find:

\(\displaystyle \L\;\;\;z^{\frac{1}{3}}\;=\;\left[2\left(\cos\frac{5\pi}{6}\,+\,i\cdot\sin\frac{5\pi}{6}\right)\right]^{\frac{1}{3}}\) ?

 

[fortwrestler]

can you please explain it step by step for me. I sent you a message, I don't know if you have read it.
Thanks

 

[stapel]

can you please explain it step by step for me.

He did explain it step by step. Which part are you stuck on?

Please reply with specifics. Thank you.

Eliz.