A community for students.
Here's the question you clicked on:
 0 viewing
Loser66
 one year ago
if \(z=e^{2\pi i/5}\) then \(1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9=??\)
Please, help
Loser66
 one year ago
if \(z=e^{2\pi i/5}\) then \(1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9=??\) Please, help

This Question is Closed

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1then \(z^4=z\\z^5=z^0=1\\z^6=z\\z^7=z^2\\z^8=z^3\\z^9=z^4=z\) so I have \(1+z+z^2+z^3+5z^4+4+4z+4z^2+4z^3+5z^4\\=5+5z+5z^2+5z^3+10z^4\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1@Michele_Laino @misty1212 @ganeshie8

Empty
 one year ago
Best ResponseYou've already chosen the best response.1Split it up into "vectors in equilibrium" and by that I mean think in terms of: \[1+z+z^2+z^3+z^4=0\] But first we can subtract 5 off of all the exponents and combine them: \(1+z+z^2+z^3+4+4z+4z^2+4z^3+5z^4\) \(5+5z+5z^2+5z^3+5z^4\) Wait a sec, we can just factor out a 5 to get: \[5(1+z+z^2+z^3+z^4)=0\]

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1No, the answer is \(5e^{3\pi i/5}\) but I don't know how to get it

Empty
 one year ago
Best ResponseYou've already chosen the best response.1Also \(z^4 \ne z\). Hmm oh I see maybe I have messed up in reading it, can you check for me that you've written it correctly as well?

Empty
 one year ago
Best ResponseYou've already chosen the best response.1I think what you were probably thinking was: \(z^4 = z^{1}\)

Empty
 one year ago
Best ResponseYou've already chosen the best response.1Ahhh ok in your original question it doesn't look like there's a \(z^4\) term thank you! Ok let me see now about this new problem.

Empty
 one year ago
Best ResponseYou've already chosen the best response.1The only difference I see is that we have a new term, \(5z^4\) so that will result in the same answer as the last one plus this. So if they are saying that the answer is \(5e^{i \frac{3 \pi}{5}}\) we can check by substituting in \(1 = e^{i \pi}\), so let's do that: \[5e^{i \frac{3 \pi}{5}} = 5e^{i \pi}e^{i \frac{3 \pi}{5}} =5 e^{i \pi \frac{3+5}{5}}=5e^{i \frac{2 \pi}{5}4}=5z^4\] Hey it works now!

Empty
 one year ago
Best ResponseYou've already chosen the best response.1If I skipped too many steps or you'd like me to explain it more, feel free to ask! :D

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1If I don't know the answer, how to derive?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1ok, I got it, hehehe but from the last sentence, I have 5(1+z+z^2+z^3+z^4) +5z^4 = 5z^4 =\(\huge5e^{8\pi i/5}= 5e^{5\pi i/5+3\pi i/5}\\\huge=5*e^{\pi i}*e^{3\pi i/5}=5e^{3\pi i/5}\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Thanks for the help. Much appreciate. I will post a new one. :) please help
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.