SOLUTION: Find the roots of : a) z3 = 1 + i b) z5 = −i c) z7 = −1

Question 1144148: Find the roots of : a) z3 = 1 + i b) z5 = −i c) z7 = −1
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Use the "^" character (shift-6) to represent exponentiation (e.g., "z^3", not just "z3"). I assume the roots to be found are for the equations

Finding roots of complex numbers is made easy with deMoivre's Theorem.

If a complex number is represented in the form

where r is a magnitude and theta is an angle of the complex number in the complex plane, then deMoivre's Theorem says that the n n-th roots of the complex number are found as follows:

(1) the magnitude of each root is the n-th root of the magnitude of the given complex number; and
(2) the "primary" n-th root has an angle that is (1/n) times the angle of the given complex number; and the other n-th roots are distributed around the circle at increments of 360/n degrees.

Here is an example....

Find the roots of

The given complex number in the required form is .

The magnitude of the given complex number is 1, so the magnitude of each of the 4-th roots is 1^(1/4)=1.

The angle of the given complex number in the complex plane is 90 degrees, so the "primary" 4-th root of i is at an angle of 90/4 = 22.5 degrees. The other 4-th roots of i are then distributed around the plane at increments of 360/4 = 90 degrees.

So the solutions to the equation are
(1)
(2)
(3)
(4)

Use the theorem to find the solutions to your examples.

To get you started -- in case you need help with this part -- the complex numbers in your examples are
(1) magnitude sqrt(2), angle 45 degrees
(2) magnitude 1, angle 90 degrees
(3) magnitude 1, angle 180 degrees

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