Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing

This Question is Closed

matrickedBest ResponseYou've already chosen the best response.1
the product of the roots αβ=1 and (α^3 β^3)=1
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
We see sum of roots and product of roots?
 one year ago

amrit110Best ResponseYou've already chosen the best response.0
\[\alpha * \beta \] isnt equal to 1....cube root remember?
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
the answer given is 0. i dont know how?
 one year ago

mayank_makBest ResponseYou've already chosen the best response.0
infact the answer is zero
 one year ago

amrit110Best ResponseYou've already chosen the best response.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. 11=0....
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
complex cube roots means α=β?
 one year ago

mayank_makBest ResponseYou've already chosen the best response.0
do u agree the roots \[\alpha =\Omega and \beta = \Omega ^{2}\]? or vice versa
 one year ago

amrit110Best ResponseYou've already chosen the best response.0
no, not necessarily all the time. for unity yes.
 one year ago

matrickedBest ResponseYou've already chosen the best response.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
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
\[ \alpha=e^{\frac{2 i \pi }{3}}; \quad \alpha^3 =1\\ \beta =1; \quad \beta^3 =1\\ \alpha \beta \ne 1 \]
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
@amrit110: can u please tell me the steps to solve this ques?
 one year ago

mayank_makBest ResponseYou've already chosen the best response.0
\[\Omega ^{3} = 1\]
 one year ago

amrit110Best ResponseYou've already chosen the best response.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.
 one year ago

shubhamsrgBest ResponseYou've already chosen the best response.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..
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
There are 3 complex roots of unity: \[ \left\{1,e^{\frac{2 i \pi }{3}},e^{\frac{2 i \pi }{3}}\right\} \]
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
A complex root of unity is a root of \[ x^3 =1 \]
 one year ago

matrickedBest ResponseYou've already chosen the best response.1
@eliassaab i didn't get how come αβ not 1
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
@amrit110: What will be the answer of (α^4 β^4) ?
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
If both \( \alpha \ne 1 \) and \( \beta \ne 1 \) then \(\alpha \beta =1\)
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
In the above case the ratio is 2.
 one year ago

amrit110Best ResponseYou've already chosen the best response.0
it should be 1 jkasdhk
 one year ago

matrickedBest ResponseYou've already chosen the best response.1
yes for complex nos .. for real nos If both α≠1 and β≠1 then αβ not 1
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
but 2 is not the answer
 one year ago

matrickedBest ResponseYou've already chosen the best response.1
is the question (α^4 β^4) 1/αβ or still different
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
If we consider complex roots only and \(\alpha \ne \beta\) then \[ \alpha = e^{\frac{2 i \pi }{3}}\\ \beta = e^{\frac{2 i \pi }{3}}\\ \alpha \beta= e^0 =1 \]
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
If the question is: Let \( \alpha, \beta\) be the 2 complex roots (not real), then the ratio in question is 2
 one year ago

jkasdhkBest ResponseYou've already chosen the best response.0
@amrit110: How is (α^4 β^4)=1
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
\[ \alpha^4 \beta^4 = \alpha^3 \beta^3 \alpha \beta = (1)(1) =\alpha \beta =1 \] If \( \alpha\ne \beta \)
 one year ago

eliassaabBest ResponseYou've already chosen the best response.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
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
This product is \[ (1)^3 \frac {1}{1}=1 \] See http://en.wikipedia.org/wiki/Vieta%27s_formulas
 one year ago
See more questions >>>
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.