Answer:
Option 2 is right
Step-by-step explanation:
Given that
[tex]z^5=1+\sqrt{3} i[/tex]
We can write this in polar form with modulus and radius
[tex]|z^5|= \sqrt{1+3} =2\\tan of Angle t =\sqrt{3} \\[/tex]
Hence angle = 60 degrees and
[tex]|z^5|= 2(cos60+isin60)[/tex]
Since we have got 5 roots for z, we can write 60, 420, 780, etc. with periods of 360
Using Demoivre theorem we get 5th root would be
5th root of 2 multiplied by 1/5 th of 60, 420, 780,....
[tex]z= \sqrt[5]{2} (cos12+isin12)\\z=\sqrt[5]{2} (cos84+isin84)\\\\z=\sqrt[5]{2} (cos156+isin156)\\\\z=\sqrt[5]{2} (cos228+isin228)\\\\z=\sqrt[5]{2} (cos300+isin300)\\[/tex]
Out of these only 2nd option suits our answer
Hence answer is Option 2.
