# Find all the seventh roots of (3 4i).

1
by Dany007

2015-10-08T15:54:34+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

To find the 7th roots of (3 + 4i), first we have to convert (3 + 4i) to polar form (r cis theta), and then set up an equation where we let z^7 = r cis (theta + k2 pi)

(where of course r is |3 + 4i| and theta is the argument (3 + 4i), or the angle between the real axis and the complex number. I take it you've done at least some of this before, so I'll move on)

We then have
z = ( r cis (theta + k2 pi))^(1/7)
= (r^(1/7)) cis ((theta/7) + (k2 pi/7)) According to De Moiré's Theorem

Sub in k = 0, 1, 2, 3, 4, 5, 6 to get all of the roots in polar form. You can then convert them back to the other form, but it's very likely to be messy.