SOLUTION: Find the four fourth roots of z=16(cos 4pi/3 + i sin 4pi/3) Write each root in standard form. Any and all help would be greatly appreciated. I don't know how to even start. Than

Click here to see ALL problems on Trigonometry-basics

Question 1170521: Find the four fourth roots of z=16(cos 4pi/3 + i sin 4pi/3) Write each root in standard form.
Any and all help would be greatly appreciated. I don't know how to even start. Thank you!
Answer by math_helper(2461) (Show Source):
You can put this solution on YOUR website!
Noting that a rotation doesn't change things, we can express z in polar form:

Now finding the four 4th roots is nearly trivial (4th root of 16 is 2, then just divide exponents by 4). You must let n=0,1,2, and 3 to get the four 4th roots:

n=0: = 2(0.5 + i*0.866) =
(or if exact answer is needed)
NOTE that is also called the principal root

Now set n=1 to find the 2nd root
n=1:
That exponent works out to :

=

Let n=2 then n=3 to get the remaining two roots. You should get:
n=2:
n=3: