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- Thread starter suvadip
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\(\displaystyle \eta = \sqrt[6]{2}\exp \left( {\frac{\pi }{{36}}} \right)\) is one sixth root of \(\displaystyle \sqrt{3}+i\).Find all the values of \(\displaystyle (\sqrt{3}+i)^{1/6}\). What is its principle value?

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.

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 \(\displaystyle \displaystyle \frac{2\pi}{6} = \frac{\pi}{3} \).