# complex number

• Apr 4th 2007, 02:43 AM
harshal54321
complex number
Hello friends,
cant find the root here!
: Find the sixth roots of (12+5i)
Thanks,
Harshal
• Apr 4th 2007, 05:24 AM
CaptainBlack
Quote:

Originally Posted by harshal54321
Hello friends,
cant find the root here!
: Find the sixth roots of (12+5i)
Thanks,
Harshal

convert the number to polar form:

12 + 5i = 13 e^(i [arctan(5/12) + 2 pi N]), N=0, +/-1. +/-2, ..

then the sixth roots are:

(12 + 5i)^(1/6) = (13)^(1/6) e^(i [arctan(5/12) + 2 pi N]/6), N=0, +/-1. +/- 2, ..

Which give distinct values for N= 0, 1, 2, 3, 4, 5.

RonL
• Apr 4th 2007, 06:25 AM
Soroban
Hello, Harshal!

I assume you're familiar with DeMoivre's Theorem . . .

Quote:

Find the sixth roots of (12 + 5i)

Convert to polar form: .12 + 5i .= .13(cosθ + i·sinθ), .where θ = arctan(5/12) .*

Then: .(12 + 5y)^
(1/6) . = . 13^(1/6) (cos[θ + 2kπ)/6] + i·sin[(θ + 2kπ)/6])

. . Evaluate for k = 0, 1, 2, 3, 4, 5

*
Edit: corrected my typo.
. . . .Thanks for pointing it out, Captain.
• Apr 4th 2007, 08:15 AM
CaptainBlack
Quote:

Originally Posted by Soroban
Hello, Harshal!

I assume you're familiar with DeMoivre's Theorem . . .

Convert to polar form: .12 + 5i .= .13(cosθ + i·sinθ), .where θ = arctan(5/13)

cos(theta)=12/13, sin(theta)=5/13, so tan(theta)=5/12

RonL