Question

theorem of nth roots

Answer 1

Using the theorem on the nth roots of a complex number, find the cube roots of the complex number in polar form. Then write the results in the form a+bi for real numbers a and b, approximated to three-decimal places. Graph the results and label completely.
\[\sqrt{2}-\sqrt{2}i\]

Answer 2

start by changing it into polar form

Answer 3

ok, let me work it out on paper. Just a moment :)

Answer 4

umm r = 2
tan(theta) = -1

Answer 5

but idk what point tan(theta) = -1 at

Answer 6

Notice that the complex number lies in IVth quadrant because the real part is positive and imaginary part is negative


so the angle would be simply 360-45 = ?

Answer 7

315

Answer 8

Yep, so the polar form is
\[\large 2\angle 315\]

Answer 9

uh the prof writes polar form as
rcis(theta)

Answer 10

\[\large 2\text{cis}(315)\]

like that ?

Answer 11

yeah:) what is 315 in radian? he likes radian xD

Answer 12

7pi/4?

Answer 13

Yep!

Answer 14

ok so now we have the polar form..what now?
2cis(7pi/4)

Answer 15

take cuberoot using demoivre's thm

Answer 16

radius remains same for all cube roots : \(\large 2^{1/3}\)

Answer 17

how many cuberoots do you expect ?

Answer 18

how does that look plugged into my polar thing? o.o

Answer 19

didn't get u

Answer 20

"my polar thing"

that sounds a bit weird lol

Answer 21

you said to do DeM's theorem
on 2cis(7pi/4)

Answer 22

yeah? haha

Answer 23

you must be knowing that there will be exactly 3 distinct cube roots for any complex number

Answer 24

you already knew the radius : \(2^{1/3}\)

just need to find the angle for each

Answer 25

Angle for each cuberoot is given by \[\large \dfrac{\frac{7\pi}{4}+2\pi \color{red}{k}}{3}\]

Answer 26

plugin \(\color{Red}{k=0}\) to get the angle of first cube root

Answer 27

= 7pi/12 ?

Answer 28

Yes, so \(\large 2^{1/3}\text{cis}(7\pi/12)\) is ur first cuberoot

Answer 29

plugin \(\color{red}{k=1}\) to get second cuberoot

Answer 30

*angle of

Answer 31

15pi/12

Answer 32

Yes, you can guess how to find the angle for third cuberoot

Answer 33

plugin \(\color{red}{k=2}\)

Answer 34

hmm how do i do the 2pi2 part?
would that become 4pi? and then multiply it so I can add it to the 7pi/4?

Answer 35

exactly!
it is actually \(2\pi*\color{red}{k}\)

Answer 36

23pi/4

Answer 37

23pi/12 right

Answer 38

hmm ok

Answer 39

so the answers would be the polar form
the 3 k's and.. what else?

Answer 40

I have to draw it on the graph

Answer 41

Yes can you list the 3 cuberoots that we just found

Answer 42

Sorry, computer spazzed. Coming.

Answer 43

\[2^{1/3}cis(7\pi/12)\] \[2^{1/3}cis(15\pi/12)\] \[2^{1/3}cis(23\pi/12)\]

Answer 44

looks good

`Then write the results in the form a+bi for real numbers a and b, approximated to three-decimal places. Graph the results and label completely.`

Answer 45

did you write the cuberoots in a+bi form ?

Answer 46

er not yet. give me a moment.

Answer 47

k

Answer 48

um.. pi/12 is not on the unit circle. how do i find cos and sin?

Answer 49

use ur calculator

Answer 50

that's probably a really dumb question but i dont know

Answer 51

oooh...okies.

Answer 52

uh k0 = 1.413 approximately. correct?

Answer 53

for cosine

Answer 54

idk il have to check wid wolfra lol

Answer 55

that's with sq2 multiplied in

Answer 56

look at decimal approximation part
http://www.wolframalpha.com/input/?i=2%5E%281%2F3%29cis%287pi%2F12%29

Answer 57

how did they get that :o

Answer 58

ooh dangit i put sq2 instead of cube rt2

Answer 59

haha okay that explains

Answer 60

I'm still getting different numbers -.-

Answer 61

work it again

Answer 62

but the answer would be
-0.326+i1.217

Answer 63

Yep, convert remaining two

Answer 64

-0.891-0.891i

Answer 65

1.217-0.326i

Answer 66

correct?

Answer 67

Looks good!

Answer 68

oo fancy!! so with just those 3 we get a circle? :o

Answer 69

radius is same for all 3 cuberoots
so they fall on the same circle

Answer 70

how did you know the shape was a circle and didn't have indents in places?

Answer 71

i just drew that circle to emphasize that radius is same for all 3 cuberoots

Answer 72

it has nothing to do with cuberoots

Answer 73

cuberoots are just the 3 numbers that we found earlier

Answer 74

haha ah ok :) just checking.

Answer 75

do you think I should draw a circle in? or just the points?

Answer 76

draw the circle also if u want to impress ur prof

Answer 77

did u notice, the 3 cuberoots are spaced out equally on that circle ?

Answer 78

lol, will do