Advertisements
Advertisements
Question
If α and β are the complex cube root of unity, show that α4 + β4 + α−1β−1 = 0
Advertisements
Solution
α and β are the complex cube roots of unity
∴ α = `(-1 + "i"sqrt(3))/2` and β = `(-1 - "i"sqrt(3))/2`
∴ αβ = `((-1 + "i"sqrt(3))/2)((-1 - "i"sqrt(3))/2)`
= `((-1)^2 - ("i"sqrt(3))^2)/4`
= `(1 - (-1)(3))/4`
= `(1 + 3)/4`
∴ αβ = 1
Also, α + β = `(-1 + "i"sqrt(3))/2 + (-1 - "i"sqrt(3))/2`
= `(-1 + "i"sqrt(3) - 1 - "i"sqrt(3))/2`
= `(-2)/2`
∴ α + β = – 1
α4 + β4 + α−1β−1
= `α^4 + β^4 + 2α^2β^2 - 2α^2β^2 + 1/"αβ"` ...[Adding and subtracting 2α2β2]
= `(α^2 + β^2)^2 - 2α^2β^2 + 1/"αβ"`
= `[(α + β)^2 - 2αβ]^2 - 2(αβ)^2 + 1/"αβ"`
= `[(-1)^2 - 2(1)]^2 - 2(1)^2 + 1/1`
= (1 – 2)2 – 2 + 1
= (– 1)2 – 1
= 1 – 1
= 0
APPEARS IN
RELATED QUESTIONS
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, find the value of `omega + 1/omega`
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 `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, then prove the following: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
Find the value of ω21
Find the value of ω–105
If ω is a complex cube root of unity, show that (1 + ω − ω2)6 = 64
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 α and β are the complex cube root of unity, show that α2 + β2 + αβ = 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| = 10
Find the equation in cartesian coordinates of the locus of z if |z – 3| = 2
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
Which of the following is the third root of `(1 + i)/sqrt2`?
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 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 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 + 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 ω is a complex cube-root of unity, then prove the following:
(ω2 + ω − 1)3 = −8
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.
(w2 + w - 1)3 = - 8
If ω is a complex cube-root of unity, then prove the following.
(ω2 + ω − 1)3 = −8
