Confirm a) & Solve b): Homework Help

[battery2004]

Homework Statement

http://img139.imageshack.us/img139/999/ksknl6.th.jpg http://g.imageshack.us/thpix.php

Could someone confirm that the a) part is correct, and if it is, then what is the next step? :)
And I am not exactly sure what to do in the b) part.

I would really appreciate if someone could help me out with this. :)

Thanks in advance.

 

[HallsofIvy]

NO, (a) is not correct because you haven't answered the question! I think you have misunderstood what is asked. If you were asked to find $^4\sqrt{16}$, you wouldn't answer "16^1/4"!

Don't change from polar to rectangular form until after you have done the root! One important reason for using polar form for complex numbers is the fact that
$$[r (cos(\theta)+ i sin(\theta)]^n= r^n(cos(n\theta)+ i sin(n\theta))$$

For n= 1/4,
$$(7(cos(\pi/2)+ i sin(\pi/2))^{1/4}= 7^{1/4}(cos(\pi/8)+ i sin(\pi/8))$$
Also, since adding $2\pi$ to the argument doesn't change the complex number, and $2\pi/4= \pi/2$ another fourth root is
$$7^{1/4}(cos(\pi/4+ \pi/2)+ i sin(\pi/4+ \pi/2))$$
yet another is
$$7^{1/4}(cos(\pi/4+ \pi)+ i sin(\pi/4+ \pi))$$
and, finally,
$$7^{1/4}(coS(\pi/4+ 3\pi/2)+ i sin(\pi/4+ 3\pi/2))$$

Those are the four fourth roots of $7(cos(\pi/2)+ i sin(\pi/2))= 7i$

 

[Nissen, Søren Rune]

Edit: Shows what I know. Listen to HallsofIvy instead

 

[m_s_a]

whith out finding all roots'
http://img155.imageshack.us/img155/1870/dddgh6.th.png