Online Cube Roots of Unity Complex Numbers Tutor & Homework Help
Solution Product of Two Imaginary Cube Roots Tutoring Online, Tutor Help
THE CUBE ROOTS OF UNITY
Let the cube root of 1 be x i.e., 3√ 1 = x.
Then by definition, x3 = 1 or x3 – 1 = 0 or (x – 1)
(x2 + x + 1) = 0
Either x – 1 = 0 i.e., x = 1 or (x2 + x + 1) = 0
Hence 
Hence, there are three cube roots of unity which are 
which the first one is real and the other two are
conjugate complex numbers. These complex cube roots of unity are also called
imaginary cube roots of unity.
which the first one is real and the other two are
conjugate complex numbers. These complex cube roots of unity are also called
imaginary cube roots of unity.
PROPERTIES OF THE CUBE ROOTS OF UNITY:
1. One imaginary cube root of unity is the square of the other.
Hence it is clear that one cube root of unity is the square of the other.
Hence if one imaginary cube root of unity be ω, then the other would be ω2.
2. The product of the two imaginary cube roots is 1.
Hence the product of the two imaginary cube roots 
(Proved)
3. The sum of the three cube roots of unity is zero.
The sum of the three cube roots of unity = 
(Proved)
Example 1:
If ω is an imaginary cube root of unity, show that (1 - ω2)(1 – ω4)
(1 – ω8)(1 – ω10) = 9
Solution:
L.H.S = (1 - ω2)(1 – ω4)(1 – ω8)(1 – ω10)
= (1 - ω2)(1 – ω)(1 - ω2)(1 – ω)
(Since ω4 = ω3. ω = 1. ω = ω; ω8 = ω6.
ω2 = 12. ω2 = ω2
and ω10 =
ω9. ω = 13. ω = ω )
= (1 - ω2)2(1 – ω )2
= (1 - 2 ω2 + ω4)
(1 - 2 ω + ω2)
= (1 + ω - 2 ω2)( 1 + ω2
- 2 ω ) (Since ω4 = ω )
= (1 + ω + ω2 - 3 ω2)
( 1 + ω + ω2 - 3 ω )
= (-3 ω2)(- 3ω )
(Since 1 + ω + ω2 = 0)
= 9 ω3 = 9.1 = 9 (Proved)
Example 2:
Factorize x2 + xy + y2.
Solution:
x2 + xy + y2 = x2 –(-1)xy + 1.y2
= x2 – ( ω + ω2)xy +
ω3y2 (Since 1 + ω + ω2 = 0 or ω +
ω2 = -1 and ω3= 1)
= x2 – xω – xyω2 +
ω3y2
= x(x – yω ) - yω 2(x – yω )
= (x – yω ) (x – yω2)
(Answer).
Online Solution Cube Roots of Unity Help:
If you are stuck with a Solution Cube Roots of Unity Homework problem and need help, we
have excellent tutors who can provide you with Homework Help. Our tutors who provide
Solution Cube Roots of Unity help are highly qualified. Our tutors have many years of
industry experience and have had years of experience providing Solution Cube Roots of
Unity Homework Help. Please do send us the Solution Cube Roots of Unity problems on
which you need Help and we will forward then to our tutors for review.
Online Tutor Product of Two Imaginary Cube Roots:
We have the best tutors in math in the industry. Our tutors can break down a complex
Product of Two Imaginary Cube Roots problem into its sub parts and explain to you in
detail how each step is performed. This approach of breaking down a problem has been
appreciated by majority of our students for learning Product of Two Imaginary Cube
Roots concepts. You will get one-to-one personalized attention through our online
tutoring which will make learning fun and easy. Our tutors are highly qualified
and hold advanced degrees. Please do send us a request for Product of Two Imaginary
Cube Roots tutoring and experience the quality yourself.
