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jkasdhk
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If α and β are complex cube roots of unity then (α^4 β^4)+1/αβ=?
 2 years ago
 2 years ago
jkasdhk Group Title
If α and β are complex cube roots of unity then (α^4 β^4)+1/αβ=?
 2 years ago
 2 years ago

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

jkasdhk Group TitleBest ResponseYou've already chosen the best response.0
How is (α^3 β^3)=1?
 2 years ago

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

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

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

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

amrit110 Group TitleBest 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....
 2 years ago

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

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

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

matricked Group TitleBest 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
 2 years ago

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

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

mayank_mak Group TitleBest ResponseYou've already chosen the best response.0
\[\Omega ^{3} = 1\]
 2 years ago

amrit110 Group TitleBest 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.
 2 years ago

shubhamsrg Group TitleBest 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..
 2 years ago

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

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

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

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

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

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

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

matricked Group TitleBest 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
 2 years ago

jkasdhk Group TitleBest ResponseYou've already chosen the best response.0
and 1/αβ would be?
 2 years ago

matricked Group TitleBest ResponseYou've already chosen the best response.1
1/αβ would be 1
 2 years ago

jkasdhk Group TitleBest ResponseYou've already chosen the best response.0
1+1 will be 2? right?
 2 years ago

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

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

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

jkasdhk Group TitleBest ResponseYou've already chosen the best response.0
its (α^4 β^4) +1/αβ
 2 years ago

eliassaab Group TitleBest 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
 2 years ago

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

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

eliassaab Group TitleBest 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
 2 years ago

eliassaab Group TitleBest 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
 2 years ago
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