Question

Which expression is a sixth root of -1+ i sqrt 3?

Answer 1

\[-1 + i \sqrt{3}\]

Answer 2

Can anyone help me do this??

Answer 3

Do you have DeMoivre? That is what you need.

Answer 4

My lesson mentions it, but they don't exactly give me a formula or anything.

Answer 5

Find the angle and magnitude represented by the number given.

Answer 6

I'm totally lost. Sorry. Can you explain to me how to do that?

Answer 7

Can you rephrase the number in Polar Coordinates. You must be able to do this or you cannot solve the problem using DeMoivre - which is, by far, the easiest way to go about it.

Map it on Cartesian coordinates and find the equivalent point in Polar Coordinates.

Answer 8

Ohh okay. Let me do that.

Answer 9

I keep getting an odd number. I don't think it's correct.

Answer 10

Show it and how you got it.

Answer 11

My schooling program just gave me answer choices as a hint:

Answer 12

Okay, how do we get the right one? You MUST be able to write the number in Polar Form. It can look like those answer choices.

Answer 13

I'm honestly not too sure how to do that.

Answer 14

Wait. Is it the z= a+bi?

Answer 15

Make sense?

Answer 16

Yes. Is the polar form x= 2(cos(\[2\pi/3) + isin (2\pi/3))\]?

Answer 17

\[z= 2(\cos(2\pi/3) + isin (2\pi/3))\]

Answer 18

Super!! Just one thing, though. Looking at the answers, we should write it in degrees.

\(x = 2\left(\cos(120º)+i\sin(120º)\right)\)

There is no compelling reason for this. It's just how the answers will want it to appear.

Okay, how do we find ALL SIX of the sixth roots of this thing?

Answer 19

Do we take the 6th root of everything?

Answer 20

This is where DeMoivre comes in.

The FIRST one is \(2^{1/6}\left(\cos(120º/6)+i\sin(120º/6)\right)\)

or \(2^{1/6}\left(\cos(20º)+i\sin(20º)\right)\)

If that's all you need to solve the problem, then we are done.
If you don't find this one in the list, you will need the other five.

Answer 21

Does this mean it would be \[\sqrt[6]{2}(\cos(20) + isin(20))\] ?

Answer 22

And there are five more. \(360º/6 = 60º\).

Every 60º there is another one. 20º, 80º, 140º, 200º, 260º, 340º

Answer 23

But how do I know which on is the sixth?

Answer 24

Nevermind. I understand now. Than you so much :)

Answer 25

The are ALL the sixth roots of the value where we started. All of them! There are 6.