# Thread: Complex Numbers: Roots of an EQN

1. ## Complex Numbers: Roots of an EQN

Hi guys, need help with this question:

Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
(1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

2. Since z = i is not a solution, I'm puzzled by the question.

3. Originally Posted by Blizzardy
Hi guys, need help with this question:

Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
(1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

Why not start by expanding $\displaystyle \displaystyle (z- i)^5$ and simplifying before trying to solve the equation?

4. Try this line of reasoning:

$\displaystyle z^{5}-(z-i)^{5}=0$

$\displaystyle z^{5}=(z-i)^{5}$

$\displaystyle \left(\frac{z-i}{z}\right)^{\!\!5}=1.$

Let

$\displaystyle w=\frac{z-i}{z}.$

Then $\displaystyle w^{5}=1,$ the fifth roots of unity (exactly). So I would try writing down the $\displaystyle w$'s, and then solving $\displaystyle zw=z-i$ for $\displaystyle z$ once you've done that.

Make sense?