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!
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.
Now, off you go. Try what you've learned.x ^ 4 = -4
equations on the set of complex numbers?
Thanks in advance!
See also this discussion at Dr. Math Forum.
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 .