Question 1144148
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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
{{{z^3 = 1+i}}}
{{{z^5 = -i}}}
{{{z^7 = -1}}}<br>
Finding roots of complex numbers is made easy with deMoivre's Theorem.<br>
If a complex number is represented in the form<br>
{{{x = r*cis(theta)}}}<br>
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:<br>
(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.<br>
Here is an example....<br>
Find the roots of {{{z^4 = i}}}<br>
The given complex number in the required form is {{{1*cis(90)}}}.<br>
The magnitude of the given complex number is 1, so the magnitude of each of the 4-th roots is 1^(1/4)=1.<br>
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.<br>
So the solutions to the equation {{{z^4=i}}} are
(1) {{{1*cis(22.5)}}}
(2) {{{1*cis(112.5)}}}
(3) {{{1*cis(202.5)}}}
(4) {{{1*cis(292.5)}}}<br>
Use the theorem to find the solutions to your examples.<br>
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