Question

find the fourth root of 16i, leave answer in rectangular form

Answer 1

please help!

Answer 2

First 4th root of 16 is 2

For the i part, put it in polar form first
\[16 i = 16 (\cos \frac{\pi}{90} + i \sin \frac{\pi}{2}) \]

then to get 4th root use this general formula
\[(\cos \theta +i \sin \theta)^{1/n} = \cos(\frac{\theta + 2 \pi k}{n}) + i \sin (\frac{\theta + 2 \pi k}{n})\]
k = 0,1,2 ... n-1

Answer 3

lol that should be pi/2 not pi/90

Answer 4

Ok, thank you for this...but what is my final answer?

Answer 5

2 (cos(pi/8)+i sin(pi/8)) = 2 ((sqrt(2+sqrt(2))/2)+(1/2 i sqrt(2-sqrt(2))))=
\[\sqrt{2+\sqrt{2}}+i\sqrt{2-\sqrt{2}}\]
I could have possibly messed up on that.

Answer 6

you are correct :)
but that is only 1 solution when k=0

you need 3 more when k=1,2,3

Answer 7

wow, thats complicated! can you please write out all four answers for me? thank you!

Answer 8

no i gave you the tools and method.... apply and find the answer on your own

Answer 9

if you just wanted answers, why not just use wolfram