Advertisements
Advertisements
Question
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
Advertisements
Solution
x = a + b, y = αa + βb and z = aβ + bα
α and β are the complex cube roots of unity.
∴ α = `(-1 + isqrt3)/2` and β = `(-1 - isqrt3)/2`
∴ αβ = `((-1 + isqrt3)/2)((-1 - isqrt3)/2)`
= `((-1)^2 - (isqrt3)^2)/4`
= `(1-(-1)(3))/4` ...[∵ i2 = -1]
= `(1 + 3)/4`
= `4/4`
∴ αβ = 1
Also, α + β = `(-1 + isqrt3)/2 + (-1 - isqrt3)/2`
= `(-1 + isqrt3 -1 - isqrt3)/2`
= `-2/2`
α + β = −1
∴ xyz = (a + b)(αa + βb)(aβ + bα)
= (a + b)(αβa2 + α2ab + β2ab + αβb2)
= (a + b)[1.(a2) + (α2 +β2)ab + 1.(b2)]
= (a + b){a2 + [(α + β)2 − 2αβ]ab + b2}
= (a + b){a2 + [(−1)2 − 2(1)]ab + b2}
= (a + b)[a2 + (1 − 2)ab + b2]
= (a + b)(a2 − ab + b2)
= a3 + b3
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 α and β are the complex cube roots of unity, show that α2 + β2 + αβ = 0.
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: (ω2 + ω - 1)3 = – 8
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 (1 + ω)3 − (1 + ω2)3 = 0
If ω is a complex cube root of unity, show that (2 + ω + ω2)3 − (1 − 3ω + ω2)3 = 65
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 α2 + β2 + αβ = 0
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
Find the equation in cartesian coordinates of the locus of z if |z − 5 + 6i| = 5
Find the equation in cartesian coordinates of the locus of z if |z – 2 – 2i| = |z + 2 + 2i|
Select the correct answer from the given alternatives:
If ω is a complex cube root of unity, then the value of ω99+ ω100 + ω101 is :
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
If (1 + ω2)m = (1 + ω4)m and ω is an imaginary cube root of unity, then least positive integral value of m is ______.
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
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+bw+cw^2))/(c+aw+bw^2) =w^2`
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 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 ω is a complex cube-root of unity, then prove the following.
(ω2 + ω − 1)3 = −8
