Advertisements
Advertisements
рдкреНрд░рд╢реНрди
If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`
Advertisements
рдЙрддреНрддрд░
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
(1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
= (1 + ω)(1 + ω2)(1 + ω)(1 + ω2) ...[тИ╡ ω3 = 1, ∴ ω4 = ω]
= (- ω2) (- ω) (- ω2) (- ω)
= ω6
= (ω3)2
= (1)2
= 1
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
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: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
Find the value of ω–30
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"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
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 − 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(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers ______.
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, 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
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
