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matricked
 2 years ago
Best ResponseYou've already chosen the best response.1the product of the roots αβ=1 and (α^3 β^3)=1

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0We see sum of roots and product of roots?

amrit110
 2 years ago
Best ResponseYou've already chosen the best response.0\[\alpha * \beta \] isnt equal to 1....cube root remember?

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0the answer given is 0. i dont know how?

mayank_mak
 2 years ago
Best ResponseYou've already chosen the best response.0infact the answer is zero

amrit110
 2 years ago
Best ResponseYou've already chosen the best response.0well 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....

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0complex cube roots means α=β?

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

amrit110
 2 years ago
Best ResponseYou've already chosen the best response.0no, not necessarily all the time. for unity yes.

matricked
 2 years ago
Best ResponseYou've already chosen the best response.1if 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

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

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0@amrit110: can u please tell me the steps to solve this ques?

amrit110
 2 years ago
Best ResponseYou've already chosen the best response.0ok. 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.

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.0note that the complex cube roots of unity are the zeroes of the eqn x^2 + x +1 so (alpha)(beta) = 1 try now..

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

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1A complex root of unity is a root of \[ x^3 =1 \]

matricked
 2 years ago
Best ResponseYou've already chosen the best response.1@eliassaab i didn't get how come αβ not 1

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0@amrit110: What will be the answer of (α^4 β^4) ?

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1If both \( \alpha \ne 1 \) and \( \beta \ne 1 \) then \(\alpha \beta =1\)

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1In the above case the ratio is 2.

amrit110
 2 years ago
Best ResponseYou've already chosen the best response.0it should be 1 jkasdhk

matricked
 2 years ago
Best ResponseYou've already chosen the best response.1yes for complex nos .. for real nos If both α≠1 and β≠1 then αβ not 1

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0but 2 is not the answer

matricked
 2 years ago
Best ResponseYou've already chosen the best response.1is the question (α^4 β^4) 1/αβ or still different

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

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

jkasdhk
 2 years ago
Best ResponseYou've already chosen the best response.0@amrit110: How is (α^4 β^4)=1

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \alpha^4 \beta^4 = \alpha^3 \beta^3 \alpha \beta = (1)(1) =\alpha \beta =1 \] If \( \alpha\ne \beta \)

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1Let \( 1, \alpha, \beta \) 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

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1This product is \[ (1)^3 \frac {1}{1}=1 \] See http://en.wikipedia.org/wiki/Vieta%27s_formulas
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