Advertisements
Advertisements
Question
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
Advertisements
Solution
Since ω is the complex cube root of unity,
ω3 = 1 and 1 + ω + ω2 = 0
∴ 1 + ω = – ω2, 1 + ω2 = – ω and ω + ω2 = – 1.
(1 − ω − ω2)3 + (1 − ω + ω2)3
= [1 – (ω + ω2)]3 + [(1 + ω2) – ω]3
= [1 – (–1)]3 + (–ω – ω)3
= 23 + (– 2ω)3
= 8 – 8ω3
= 8 – 8(1)
= 0
APPEARS IN
RELATED QUESTIONS
If ω is a complex cube root of unity, find the value of (1 - ω - ω2)3 + (1 - ω + ω2)3
If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`
If x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a3 + b3.
If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
If ω is a complex cube root of unity, show that (a + b) + (aω + bω2) + (aω2 + bω) = 0
If ω is a complex cube root of unity, show that (a − b) (a − bω) (a − bω2) = a3 − b3
If ω is a complex cube root of unity, show that (a + b)2 + (aω + bω2)2 + (aω2 + bω)2 = 6ab
If ω is a complex cube root of unity, find the value of `ω + 1/ω`
If ω is a complex cube root of unity, find the value of ω2 + ω3 + ω4
If ω is a complex cube root of unity, find the value of (1 + ω2)3
If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
If α and β are the complex cube root of unity, show that α4 + β4 + α−1β−1 = 0
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
If ω is the cube root of unity then find the value of `((-1 + "i"sqrt(3))/2)^18 + ((-1 - "i"sqrt(3))/2)^18`
The value of the expression 1.(2 – ω) + (2 – ω2) + 2.(3 – ω)(3 – ω2) + ....... + (n – 1)(n – ω)(n – ω2), where ω is an imaginary cube root of unity is ______.
If the cube roots of the unity are 1, ω and ω2, then the roots of the equation (x – 1)3 + 8 = 0, are ______.
If w is a complex cube root of unity, show that
`((a + bw + cw^2)) /( c + aw + bw^2 )= w^2`
If w is a complex cube root of unity, show that, `((a + bw + cw^2))/(c + aw + bw^2) = w^2`
If w is a complex cube root of unity, show that `((a + bw +cw^2))/(c +aw + bw^2) = w^2`
If w is a complex cube-root of unity, then prove the following:
(ω2 + ω − 1)3 = −8
If ω is a complex cube root of unity, then prove the following.
(ω2 + ω −1)3 = −8
If ω is a complex cube-root of unity, then prove the following:
(a + b) + (aω + bω2) + (aω2 + bω) = 0
If w is a complex cube root of unity, show that `((a + bω + cω^2))/(c + aω + bω^2) = ω^2`
If ω is a complex cube-root of unity, then prove the following :
(ω2 + ω − 1)3 = − 8
Find the value of `sqrt(-3) xx sqrt(-6)`.
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2)=w^2`
If w is a complex cube root of unity, show that `((a + bomega + comega^2))/(c + aomega + bomega^2) = w^2`
If ω is a complex cube root of unity, show that `((a + b\omega + c\omega^2))/(c + a\omega + b\omega^2) = \omega^2`
If ω is a complex cube-root of unity, then prove the following.
(ω2 + ω − 1)3 = −8
