Question

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

Answer 1

is it 2

Answer 2

How?

Answer 3

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

Answer 4

How is (α^3 β^3)=1?

Answer 5

We see sum of roots and product of roots?

Answer 6

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

Answer 7

the answer given is 0. i dont know how?

Answer 8

infact the answer is zero

Answer 9

How????

Answer 10

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....

Answer 11

complex cube roots means α=β?

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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.

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

In the above case the ratio is 2.

Answer 26

it should be 1 jkasdhk

Answer 27

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

Answer 28

and 1/αβ would be?

Answer 29

1/αβ would be 1

Answer 30

1+1 will be 2? right?

Answer 31

but 2 is not the answer

Answer 32

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

Answer 33

If we consider complex roots only and \(\alpha \ne \beta\) then

\[
\alpha = e^{\frac{2 i \pi
}{3}}\\
\beta = e^{-\frac{2 i \pi
}{3}}\\
\alpha \beta= e^0 =1
\]

Answer 34

its (α^4 β^4) +1/αβ

Answer 35

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

Answer 36

@amrit110: How is (α^4 β^4)=1

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

@jkasdhk