Question

Which of the following is a solution of z^5 = 1 + √3 i?

Answer 1

Answer:

Option 2 is right

Step-by-step explanation:

Given that

$z^5=1+\sqrt{3} i$

We can write this in polar form with modulus and radius

$|z^5|= \sqrt{1+3} =2\\tan of Angle t =\sqrt{3}$

Hence angle = 60 degrees and

$|z^5|= 2(cos60+isin60)$

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,....

$z= \sqrt[5]{2} (cos12+isin12)\\z=\sqrt[5]{2} (cos84+isin84)\\\\z=\sqrt[5]{2} (cos156+isin156)\\\\z=\sqrt[5]{2} (cos228+isin228)\\\\z=\sqrt[5]{2} (cos300+isin300)$

Out of these only 2nd option suits our answer

Hence answer is Option 2.