SOLUTION: What are the two square roots of -32 + 32 sqrt 3i

Algebra ->  Trigonometry-basics -> SOLUTION: What are the two square roots of -32 + 32 sqrt 3i      Log On


   

Question 983181: What are the two square roots of -32 + 32 sqrt 3i
Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!

-32+%2B+32sqrt%283%29%2Ai = 64%2A%28-%281%2F2%29+%2B+%28sqrt%283%29%2F2%29%2Ai%29 = 64%2A%28cos%282pi%2F3%29+%2B+i%2Asin%282pi%2F3%29%29.

Therefore, the two square roots of this complex number are

8%2A%28cos%28pi%2F3%29+%2B+i%2Asin%28pi%2F3%29%29 = 8*(cos(60°) + i*sin(60°)) = 8%2A%28%281%2F2%29+%2B+i%2A%28sqrt%283%29%2F2%29%29

and

8%2A%28cos%28pi%2F3+%2B+pi%29+%2B+i%2Asin%28pi%2F3+%2B+pi%29%29 = 8*(cos(60°+ 180°) + i*sin(60°+180°)) = 8*(cos(240°) + i*sin(240°)) = 8%2A%28%28-1%2F2%29+-+i%2A%28sqrt%283%29%2F2%29%29 = -8%2A%28%281%2F2%29+%2B+i%2A%28sqrt%283%29%2F2%29%29.

See my lessons on complex numbers in this site  REVIEW of lessons on complex numbers  and especially the lesson  How to take a root of a complex number.