Principal value of complex number

[Suvadip]

Find all the values of \(\displaystyle (\sqrt{3}+i)^{1/6}\). What is its principle value?I have doubt about the second part. We have heard about the principal value of the amplitude of a complex number. But here the principal value of the complex number itself is asked for. Please help

 

[Plato2]

Re: principal value of complex number

Find all the values of \(\displaystyle (\sqrt{3}+i)^{1/6}\). What is its principle value?

\(\displaystyle \eta = \sqrt[6]{2}\exp \left( {\frac{\pi }{{36}}} \right)\) is one sixth root of \(\displaystyle \sqrt{3}+i\).

If \(\displaystyle \zeta =\exp \left( {\frac{\pi }{{3}}} \right)\) then \(\displaystyle \eta\cdot\zeta^k,~k=0,1,\cdots 5\) are all six.

I have seen \(\displaystyle \eta\) (i.e. \(\displaystyle k=0\)) called the principal root.

 

[Prove It]

Find all the values of \(\displaystyle (\sqrt{3}+i)^{1/6}\). What is its principle value?I have doubt about the second part. We have heard about the principal value of the amplitude of a complex number. But here the principal value of the complex number itself is asked for. Please help

It helps if you remember that there are always two square roots, three cube roots, four fourth roots, etc, and they are all evenly spaced around a circle. So in this case, if you can evaluate one value, the rest will all have the same magnitude and be separated by an angle of \(\displaystyle \displaystyle \frac{2\pi}{6} = \frac{\pi}{3} \).