SOLUTION: I need help with: Write the three cube roots of i. Express the roots in rectangular form. Use exact values.

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Question 982703: I need help with: Write the three cube roots of i. Express the roots in rectangular form. Use exact values.
Found 2 solutions by ikleyn, solver91311:
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!

root%283%2C+i%29  has  3  values:

1)  cos%28%281%2F3%29%2A%28pi%2F2%29%29+%2B+i%2Asin%28%281%2F3%29%2A%28pi%2F2%29%29 = cos%28pi%2F6%29+%2B+i%2Asin%28pi%2F6%29 = cos(30°) + i*sin(30°) = sqrt%283%29%2F2+%2B+%281%2F2%29%2Ai;

2)  cos%28pi%2F6++%2B+2pi%2F3%29+%2B+i%2Asin%28pi%2F6+%2B+2pi%2F3%29 = cos(30°+120°) + i*sin(30° + 120°) = cos(150°) + i*sin(150°) = -sqrt%283%29%2F2+%2B+%281%2F2%29%2Ai;

3)  cos%28pi%2F6++%2B+2%2A%282pi%2F3%29%29+%2B+i%2Asin%28pi%2F6+%2B+2%2A%282pi%2F3%29%29 = cos(30°+240°) + i*sin(30° + 240°) = cos(270°) + i*sin(270°) = 0+-+1%2Ai = -i.


Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


While ikleyn's solution using De Movire's Theorem has more general application, here is another method that is more specific to this particular problem that has the advantage of simple elegance:

Consider that . Then we can say:



Next, recall the factorization of the difference of two cubes:



which simplifies to:



So, if



then



but if



then



John

My calculator said it, I believe it, that settles it