Step by step solution to "How can you find the complex roots of..."

Answered question

2022-01-17

How can you find the complex roots of i?
A variation of the Root of Unity problem.
I want to find all possible answers to this:

z^{n} = i

Where

i^{2} = - 1

nick1337

Expert2022-01-19Added 777 answers

Step 1 If the polar form of z is

z = r (\cos θ + i \sin θ)

there are n distinct solutions to the equation

w^{n} = z

w = \sqrt[n]{r} (\cos \frac{θ + 2 π k}{n} + i \sin \frac{θ + 2 π k}{n})

where

k = 0, 1, \dots, n - 1

. In your case,

z = i

, whose polar form is given by

r = 1, θ = \frac{π}{2}

Vasquez

Expert2022-01-19Added 669 answers

Step 1 Also, observe that if

z^{n} = i

then

z^{4 n} = 1

. Thus, the complex numbers youre

alenahelenash

Expert2022-01-24Added 556 answers

Generally, the answers would be of the form

\sqrt[n]{i} w_{n}^{j}

where

w_{n} = \exp (\frac{2 i π}{n})

is a root of unity, and

j = 0 \dots n - 1

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?