Advertisements
Advertisements
Question
If ω is a complex cube root of unity, find the value of ω2 + ω3 + ω4.
Advertisements
Solution
ω is a complex cube root of unity
∴ ω3 = 1 and 1 + ω + ω2 = 0
Also, 1 + ω2 = - ω, 1 + ω = - ω2 and ω + ω2 = – 1
ω2 + ω3 + ω4
= ω2 (1 + ω + ω2) = ω2(0) = 0
APPEARS IN
RELATED QUESTIONS
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: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
Find the value of ω–105
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
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 (a + b) + (aω + bω2) + (aω2 + bω) = 0
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 ω2 + ω3 + ω4
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
If α and β are the complex cube root of unity, show that α4 + β4 + α−1β−1 = 0
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" + 3"i")/("z" - 6"i")|` = 1
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`
Which of the following is the third root of `(1 + i)/sqrt2`?
Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 – α3033 is equal to ______.
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`
