Advertisements
Advertisements
Question
If ω is a complex cube root of unity, show that (2 + ω + ω2)3 - (1 - 3ω + ω2)3 = 65
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
L.H.S. = (2 + ω + ω2)3 - (1 - 3ω + ω2)3
= [(2 + (ω + ω2)]3 - [(- 3ω + (1 + ω2)]3
= (2 - 1)3 - (- 3ω - ω)3
= 13 - (- 4ω)3
= 1 + 64ω3
= 1 + 64(1) = 65
= R.H.S.
APPEARS IN
RELATED QUESTIONS
If ω is a complex cube root of unity, find the value of `omega + 1/omega`
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 (2 − ω)(2 − ω2) = 7
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) (a − bω) (a − bω2) = a3 − b3
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
Which of the following is the third root of `(1 + i)/sqrt2`?
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 ω is a complex cube-root of unity, then prove the following:
(a + b) + (aω + bω2) + (aω2 + bω) = 0
If ω is a complex cube-root of unity, then prove the following :
(ω2 + ω − 1)3 = − 8
