Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

jkasdhk

  • 2 years ago

If α and β are complex cube roots of unity then (α^4 β^4)+1/αβ=?

  • This Question is Closed
  1. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    is it 2

  2. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How?

  3. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    the product of the roots αβ=1 and (α^3 β^3)=1

  4. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How is (α^3 β^3)=1?

  5. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    We see sum of roots and product of roots?

  6. amrit110
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\alpha * \beta \] isnt equal to 1....cube root remember?

  7. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the answer given is 0. i dont know how?

  8. mayank_mak
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    infact the answer is zero

  9. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How????

  10. amrit110
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    well its complex cube root, \[\alpha = \beta \] = \[\sqrt{-1}\]. so that would make cube of it equal to 1. but the other term -1. 1-1=0....

  11. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    complex cube roots means α=β?

  12. mayank_mak
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do u agree the roots \[\alpha =\Omega and \beta = \Omega ^{2}\]? or vice versa

  13. amrit110
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no, not necessarily all the time. for unity yes.

  14. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if α and β are the complex cube roots of 1 then α and β must satisfy the eq. x^2 +x +1=0 and x^3=1 thus α + β=-1 and αβ=1

  15. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[ \alpha=e^{\frac{2 i \pi }{3}}; \quad \alpha^3 =1\\ \beta =1; \quad \beta^3 =1\\ \alpha \beta \ne 1 \]

  16. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @amrit110: can u please tell me the steps to solve this ques?

  17. mayank_mak
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\Omega ^{3} = 1\]

  18. amrit110
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok. see complex cube roots of unity so \[x^{3}=1\]. so when u solve for x it is obviously the root of (-1). now this means that for this case we can safely assume alpha n beta to be equal to x. although this is not true for all cubic equations, sometimes the roots can be complex conjugate too. but it doesnt matter.

  19. shubhamsrg
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    note that the complex cube roots of unity are the zeroes of the eqn x^2 + x +1 so (alpha)(beta) = 1 try now..

  20. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    There are 3 complex roots of unity: \[ \left\{1,e^{\frac{2 i \pi }{3}},e^{-\frac{2 i \pi }{3}}\right\} \]

  21. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    A complex root of unity is a root of \[ x^3 =1 \]

  22. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @eliassaab i didn't get how come αβ not 1

  23. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @amrit110: What will be the answer of (α^4 β^4) ?

  24. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If both \( \alpha \ne 1 \) and \( \beta \ne 1 \) then \(\alpha \beta =1\)

  25. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    In the above case the ratio is 2.

  26. amrit110
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it should be 1 jkasdhk

  27. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes for complex nos .. for real nos If both α≠1 and β≠1 then αβ not 1

  28. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and 1/αβ would be?

  29. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    1/αβ would be 1

  30. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    1+1 will be 2? right?

  31. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but 2 is not the answer

  32. matricked
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    is the question (α^4 β^4) -1/αβ or still different

  33. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If we consider complex roots only and \(\alpha \ne \beta\) then \[ \alpha = e^{\frac{2 i \pi }{3}}\\ \beta = e^{-\frac{2 i \pi }{3}}\\ \alpha \beta= e^0 =1 \]

  34. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    its (α^4 β^4) +1/αβ

  35. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If the question is: Let \( \alpha, \beta\) be the 2 complex roots (not real), then the ratio in question is 2

  36. jkasdhk
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @amrit110: How is (α^4 β^4)=1

  37. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[ \alpha^4 \beta^4 = \alpha^3 \beta^3 \alpha \beta = (1)(1) =\alpha \beta =1 \] If \( \alpha\ne \beta \)

  38. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Let \( 1, \alpha, \beta \) be the three roots of unity,then \[ \frac { \alpha ^4 \beta^4 +1}{\alpha \beta }= 2 \] It is easy to see now \[ (1) \alpha \beta =1 \] Since the last product is the product of the three roots of the polynomial \[ x^3 -1=0

  39. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    This product is \[ (-1)^3 \frac {-1}{1}=1 \] See http://en.wikipedia.org/wiki/Vieta%27s_formulas

  40. eliassaab
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @jkasdhk

  41. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find 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
  • 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.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.