# Complex Numbers: Roots of an EQN

• Jun 2nd 2011, 06:54 PM
Blizzardy
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.

• Jun 2nd 2011, 07:42 PM
TKHunny
Since z = i is not a solution, I'm puzzled by the question.
• Jun 2nd 2011, 08:55 PM
Prove It
Quote:

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?
• Jun 3rd 2011, 02:06 AM
Ackbeet
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?