Cube Root of Unity
2026-02-28 06:06 Diff
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Last updated on August 5, 2025

The cube root of Unity (or one) are the values which, when multiplied together, gives the original number Unity. The Cube Root of Unity is represented by ∛1, which actually have three roots→ 1,𝛚, 𝛚², which on multiplication together gives “1” as a product. 1×𝛚×𝛚²=1.

What Is the Cube Root of Unity?

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As mentioned above, the cube root of Unity are 1,𝛚, 𝛚², where 1 is a real root, 𝛚 and 𝛚² are the imaginary roots.
The essential features or properties of the cube root of Unity are:


The imaginary roots 𝛚 and 𝛚², when multiplied together, yields 1


𝛚×𝛚²= 𝛚³=1


The summation of the roots is zero → 1+𝛚+𝛚²=0.


The imaginary root 𝛚, when squared, is expressed as 𝛚², which is equal to another imaginary root.


Fact check: Do you know? The values of Cube root of (-1) are -1, -𝛚, and -𝛚² 

Finding the Cube Root of Unity

Now, let us find the meaning of 𝛚 here. To find the cube root of Unity, we will make use of some algebraic formulas. We know that, the cube root of unity is represented as ∛1. Let us assume that ∛1= a so,


∛1= a


⇒ 1 = a3


⇒ a3- 1 = 0 


 ⇒ (a - 1)(a2+a+1) = 0              [using a3-b3= (a - b)(a2+a.b+b2)]


⇒a - 1 =0


⇒ a= 1  …………..(1) 


Again, a2+a+1 = 0


    ⇒ a = (-1 ±√(12–4×1×1)) / 2×1


⇒ a =  (-1 ±√(–3)) / 2


⇒ a =  (-1 ± i√3) / 2


⇒ a =   (-1 + i√3) / 2     …………(2)


Or


a =  (-1 - i√3) / 2        …………(3)


From equation (1), (2), and (3), we get,


The roots are →  1,  (-1 + i√3) / 2 and  (-1 - i√3) / 2


 Hence, 𝛚 = (-1 + i√3) / 2


𝛚2= (-1 - i√3) / 2
 

Common Mistakes and How to Avoid Them in the Cube Root of Unity

some common mistakes with their solutions given:

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Problem 1

Factorize m²+ mn + n²

Okay, lets begin

We know that, 1+𝛚+𝛚2=0


⇒ 𝛚+𝛚2= -1 ……….(1)


And, 𝛚3=1 …….(2)


So, m2+mn+n2


= m2 - (-1)mn +1× n2


= m2 - (𝛚+𝛚2)mn + 𝛚3× n2       [Using (1) and (2)]


= m2- mn𝛚- mn𝛚2+ n2𝛚3


= m(m-n𝛚) -n𝛚2(m-n𝛚)


= (m-n𝛚)(m-n𝛚2)


Answer : (m-n𝛚)(m-n𝛚2)
 

Explanation

We used the properties of the cube root of unity to factorise the expression.
 

Well explained 👍

Problem 2

Find 𝛚⁶⁶

Okay, lets begin

𝛚66

=(𝛚3)22

=(1)22

=1


Answer: 1
 

Explanation

We used the property 𝛚3=1, and solved the expression.
 

Well explained 👍

Problem 3

Prove that (1+𝛚)³+(1+𝛚²)³ = -2

Okay, lets begin

We know that, 1+𝛚+𝛚2=0
            ⇒1+𝛚= -𝛚2 ……….(1)
And also, 1+𝛚2= -𝛚 ………(2)

   LHS = (1+𝛚)3+(1+𝛚2)3


=(-𝛚2)3+(-𝛚)3      [Using (1) and (2)]


=(-𝛚6)+(-𝛚3) 


= -(𝛚3)2 - (𝛚3)


=-(1)2 - 1              [using the property 𝛚3=1]


= -1-1


=-2


=RHS [proved]
 

Explanation

We proved the given expression to be true using properties of cube root if unity like 1+𝛚+𝛚2=0 and 𝛚3=1.
 

Well explained 👍

Problem 4

Prove that (1+𝛚-𝛚²)⁶= -64

Okay, lets begin

We know that, 1+𝛚+𝛚2=0


 ⇒1+ 𝛚= -𝛚2 ……….(1)


And, 𝛚3=1 …….(2)


LHS


= (1+𝛚-𝛚2)6


=(-𝛚2-𝛚2)6    [using (1)]


=(-2𝛚2)6


=26 × (-𝛚2)6


=64× (-𝛚12)


= 64× (-(𝛚3)4)


= 64× (-(1)4)


= 64× (-1)


= -64


=RHS
 

Explanation

LHS=RHS

Hence proved

Well explained 👍

FAQs for Cube Root of Unity

1.What is the cube root of unity rule?

 The cube root of unity has three roots → 1,𝛚, and 𝛚2, 1 is the real root, and 𝛚 and 𝛚2 are the imaginary roots. Also, the important properties of cube root of unity include: 


1+𝛚+𝛚2=0


𝛚3=1


The square of the imaginary root omega, 𝛚 is the other imaginary root, 𝛚2
 

2.What is the expression of the cube root of unity ?

The expression of cube root of unity is x3=1 or, x=∛1. We generally solve this expression by using the formula a3-b3= (a - b)(a2+a.b+b2). 
 

3.How do you find the cube root of unity?

We can find the cube root of unity, i.e., the values of 𝛚 and 𝛚2 by solving the expression x3=1 or, x=∛1, using the formula a3-b3= (a - b)(a2+a.b+b2). We then land on to a value which deals with the Complex numbers

4. Is the cube root of unity 1?

The cube root of unity is not only 1, it comprises two imaginary roots also, 


𝛚 = (-1 + i√3) / 2 and 𝛚2= (-1 - i√3) / 2.
 

Important Glossaries for Cube Root of Unity

  • Omega - This is a symbol used for depicting the imaginary roots of the cube root of unity. It is represented by 𝛚. 
  • Complex Number - The numbers which are represented as m+i.n, where m and n are real numbers and “i”, known as iota, is an imaginary number. 
     

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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: He loves to play the quiz with kids through algebra to make kids love it.