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Last updated on May 26th, 2025

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Cube Root of Unity

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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 βˆ›1, which actually have three rootsβ†’ 1,π›š, π›šΒ², which on multiplication together gives β€œ1” as a product. 1Γ—π›šΓ—π›šΒ²=1.

Cube Root of Unity for Thai Students
Professor Greenline from BrightChamps

What Is the Cube Root of Unity?

As mentioned above, the cube root of Unity are 1,π›š, π›š², where 1 is a real root, π›š and π›š² are the imaginary roots.
The essential features or properties of the cube root of Unity are:


The imaginary roots π›š and π›š², when multiplied together, yields 1


π›š×π›š²= π›š³=1


The summation of the roots is zero → 1+π›š+π›š²=0.


The imaginary root π›š, when squared, is expressed as π›š², which is equal to another imaginary root.


Fact check: Do you know? The values of Cube root of (-1) are -1, -π›š, and -π›š² 

Professor Greenline from BrightChamps

Finding the Cube Root of Unity

Now, let us find the meaning of π›š 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 βˆ›1. Let us assume that βˆ›1= a so,


βˆ›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, π›š = (-1 + i√3) / 2


π›š2= (-1 - i√3) / 2
 

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

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Cube Root of Unity Examples

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Problem 1

Factorize mΒ²+ mn + nΒ²

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Explanation

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Problem 2

Find π›šβΆβΆ

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Explanation

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Problem 3

Prove that (1+π›š)Β³+(1+π›šΒ²)Β³ = -2

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Explanation

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Problem 4

Prove that (1+π›š-π›šΒ²)⁢= -64

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Explanation

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FAQs for Cube Root of Unity

1.What is the cube root of unity rule?

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

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

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4. Is the cube root of unity 1?

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5.How does learning Algebra help students in Thailand make better decisions in daily life?

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6.How can cultural or local activities in Thailand support learning Algebra topics such as Cube Root of Unity?

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7.How do technology and digital tools in Thailand support learning Algebra and Cube Root of Unity?

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8.Does learning Algebra support future career opportunities for students in Thailand?

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Professor Greenline from BrightChamps

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 π›š. 

 

  • 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. 
     
Professor Greenline from BrightChamps

About BrightChamps in Thailand

At BrightChamps, we understand algebra is much more than symbolsβ€”it’s a way to unlock endless opportunities! We help children throughout Thailand grasp important math concepts, focusing today on the Cube Root of Unity with a special emphasis on cube rootsβ€”in an engaging, enjoyable, and easy-to-understand manner. Whether your child is calculating the speed of a roller coaster at Dream World, keeping score at a local football match, or managing their allowance to buy the latest gadgets, mastering algebra builds the confidence they need in daily life. Our interactive lessons are designed to be both fun and simple. Knowing kids in Thailand learn in diverse ways, we personalize our teaching for each learner. From the bustling streets of Bangkok to the tropical islands of Phuket, BrightChamps makes math relevant and exciting across Thailand. Let’s bring cube roots into every child’s math story!
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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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Fun Fact

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

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