Question

Find the fourth roots of 256(cos 240° + i sin 240°).

Answer 1

@Loser66

Answer 2

@whpalmer4
Please anybody

Answer 3

Had this same question up for 30 min and nobody replied :(

Answer 4

do you know De Moivre's Theorem?

Answer 5

No but i can pull it up real quick if i need it

Answer 6

[r(costheta + isintheta)] right?

Answer 7

= r^n(cos ntheta + isin n theta) that the formula.

Answer 8

take a look at Demoivre theorem at http://www.intmath.com/complex-numbers/7-powers-roots-demoivre.php

Answer 9

Are you supposed to take the 4th power, or the 4th root?

Answer 10

OK. Am i supposed to simplify the equation that u gave me? And the only information they give me is the fourth roots of it. Idk about if it wants powers

Answer 11

take the fourth root of 256 and divide the angle by 4
that is all

Answer 12

do as satellite73 says, kjuchiha.

Answer 13

fourth root of 256 = 4, divided by 4 = 1
That cant be it, can it?

Answer 14

the fourth root of 256 is 4
\(240\div 4=60\)

Answer 15

no, don't divide the fourth root of 256 by 4, divide the angle by 4

Answer 16

4(cos60 + i*sin60)?

Answer 17

yes

Answer 18

wasn't that easy?

Answer 19

Holy... thank you! So much easier than I thought! took an hour of me not being able to figure it out

Answer 20

Thanks satellite and all others!

Answer 21

well, interesting that Mathematica gets a different answer...

Answer 22

Hmm... is that bad?

Answer 23

now if you want to be fancy, you can say \(\cos(60)=\frac{1}{2}\) and \(\sin(60)=\frac{\sqrt{3}}{2}\) and so the FINAL ANSWER would be \[4\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)\]\[=2+2\sqrt{3}i\]

Answer 24

http://www.wolframalpha.com/input/?i=(256+(cos(240+degrees)+%2B+i+sin(240+degrees)))%5E(1%2F4)

Answer 25

don't forget that there are not one, but 4 fourth roots

Answer 26

What does that mean satellite? there's more?

Answer 27

we found one of them, lets find another

Answer 28

@satellite73 so, if I apply Demoivre theorem with power of 1/4 , I get the same answer, right?

Answer 29

the number out front will stay at 4, but we can go around the circle one more time and say \(240+360=600\) and \(600\div 4=150\)

Answer 30

@Loser66 yes, this is de moivre

Answer 31

bingo.

Answer 32

360/4 = 90, so shouldnt I just add 90 to each angle?
4(cos 150 + i sin 150)
4(cos 240 + i sin 240)
4(cos 330 + i sin 330)

Answer 33

so another answer is
\[4\left(\cos(150)+i\sin(150)\right)\]

Answer 34

So all in all its
4(cos 60 + i sin 60)
4(cos 150 + i sin 150)
4(cos 240 + i sin 240)
4(cos 330 + i sin 330)

Answer 35

yes, add 90 works as well

Answer 36

you found the sound

Answer 37

Wow, thats too easy!

Answer 38

yes, that is kind of the point of demoivre
it is easy not hard
otherwise, why would you write a number in the form \(a+bi\) as
\[r\left(\cos(\theta)+i\sin(\theta)\right)\]?

Answer 39

thank you!

Answer 40

yw

Answer 41

Good point about the additional roots — I was thrown off by the after that I only got one when I evaluated it, and normally I get multiple roots from Mathematica. Difference was that I hadn't asked it to solve this time, so it didn't present all the candidates.

Answer 42

i wonder what happens if we do this

Answer 43

http://www.wolframalpha.com/input/?i=%28%28256%28cos%28240%29%2Bisin%28240%29%29%29^%281%2F4%29

Answer 44

\[\{\{x\to -3.4641+2. i\},\{x\to -2.-3.4641 i\},\]\[\{x\to 2.\, +3.4641 i\},\{x\to 3.4641\, -2. i\}\}\]

Answer 45

WA does the same if we ask it to solve x^4 = 256cis(240 deg)

http://www.wolframalpha.com/input/?i=%28256%28cos%28240%29%2Bisin%28240%29%29%29%3Dx%5E4