Question

Math Analysis:
Find the indicated roots and express in both polar and rectangular form.Exact answers please!
15. Fourth roots of 8sqr. rt.3 + 8i

Answer 1

\[8\sqrt{3} + 8i\]

Answer 2

\[
\tan (\theta) = \frac {8}{8\sqrt 3}=\frac {1}{\sqrt 3}\\
\theta = \frac\pi 6\\
r=\sqrt { 8^2 + (8 \sqrt 3)^2 }=\sqrt{256}=16
\]
What are the polar coordinates?

Answer 3

how did yo get pi/6?

Answer 4

oh alrite i get it :)

Answer 5

what do you mean the polar coordinates?

Answer 6

it would be 16cis pi/6 rite?

Answer 7

or 16cis30

Answer 8

\[
8\sqrt 3 + 8 i = 16 e^{ i \frac \pi 6}
\]
How can you write the 4th root of that in polar?

Answer 9

\[
(16 e^{ i \frac \pi 6})^\frac 1{ 4} = 2 e^{\frac{i \pi }{24}}
\]

Answer 10

this is what my teacher wrote
(16cis30)^1/4
2cis(7.5+90k) , k=0,1,2,3

Answer 11

k=0: 2cis7.5
k=1: 2cis97.5
k=2: 2cis187.5
k=3: 2cis277.5

Answer 12

You gave the general answer. That is great.

Answer 13

i still dont get it thow i just wrote down what my teacher wrote on the board xD

Answer 14

how did he get 7.5 in 2cis(7.5+90k)?

Answer 15

oh wait i get it he divided 30 by 4 :)

Answer 16

but why would i add 7.5 by 90

Answer 17

in 2cis(7.5+90k)

Answer 18

Every non zero complex number has n n_th roots.

\[
(8\sqrt 3 + 8 i)^{\frac 1 4} = \left(16 e^{ i \left(\frac \pi 6 + 2 k \pi\right)}\right)^{\frac 1 4}= 2 e^{ i \left(\frac \pi {24} + k \frac \pi 2\right)}, \, k=0, 1, 2 ,3
\]

Answer 19

Did you get it?

Answer 20

so that's why i add it by 90?