Hello,

Can someone help, how to solve

x ^ 3 = 1

and/or

x ^ 4 = -4

equations on the set of complex numbers?

Thanks in advance!

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- October 31st 2010, 10:44 AMFujiwaraEquations on the set of complex numbers
Hello,

Can someone help, how to solve

x ^ 3 = 1

and/or

x ^ 4 = -4

equations on the set of complex numbers?

Thanks in advance! - October 31st 2010, 11:16 AMJhevon
I'll do the first, I leave figuring out the second to youWe will solve equation in polar form, and then we can convert to the more familiar form using Euler's equation,

Let , then we have

But by Euler's equation, , hence

, and

for (after that, the solutions repeat)

Hence,

Which, by Euler's equation, gives:

those are the three roots in the complex numbers.

Quote:

x ^ 4 = -4

equations on the set of complex numbers?

Thanks in advance!

- October 31st 2010, 11:23 AMemakarov
See also this discussion at Dr. Math Forum.

- November 1st 2010, 12:49 AMFujiwara
Thanks for your help, now I understand and I can solve the problem.

- November 1st 2010, 08:08 AMHallsofIvy
Another way to solve , which is equivalent to is to factor it as . You can then solve with the quadratic formula.

is the same as and you can think of that as the difference of two squares: with and . . But you will now need to change to polar form to find and .