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Last updated on November 30th, 2024

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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 Qatari 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 Qatar make better decisions in daily life?

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

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

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

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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 Qatar

At BrightChamps, we believe algebra is more than just numbersβ€”it opens doors to endless opportunities! We’re here to help children across Qatar master important math skills, focusing today on the Cube Root of Unity with a special focus on cube rootsβ€”in an engaging, enjoyable, and easy-to-understand way. Whether your child is calculating the speed of a roller coaster at Qatar’s Angry Birds World, tracking scores at a local football match, or managing their allowance for the latest gadgets, mastering algebra builds confidence for everyday life. Our interactive lessons are designed to make learning fun and simple. Knowing kids in Qatar learn in diverse ways, we personalize our teaching for each child. From the modern skyline of Doha to the vast desert landscapes, BrightChamps brings math to life, making it exciting across Qatar. Let’s make cube roots a fun part of every child’s math journey!
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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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