285 LearnersLast updated on May 26th, 2025
Cube Root of Unity
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 β1, which actually have three rootsβ 1,π, πΒ², which on multiplication together gives β1β as a product. 1ΓπΓπΒ²=1.
What Is the Cube Root of Unity?
As mentioned above, the cube root of Unity are 1,π, π², where 1 is a real root, π and π² are the imaginary roots.
The essential features or properties of the cube root of Unity are:
The imaginary roots π and π², when multiplied together, yields 1
π×π²= π³=1
The summation of the roots is zero → 1+π+π²=0.
The imaginary root π, when squared, is expressed as π², which is equal to another imaginary root.
Fact check: Do you know? The values of Cube root of (-1) are -1, -π, and -π²
Finding the Cube Root of Unity
Now, let us find the meaning of π 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 β1. Let us assume that β1= a so,
β1= 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 = (-1 + i√3) / 2
π2= (-1 - i√3) / 2
Common Mistakes and How to Avoid Them in the Cube Root of Unity
some common mistakes with their solutions given:
Mistake 1
Students often do 1+π+π2=1
It should be clear to students that, 1+π+π2=0 and π3=1
Cube Root of Unity Examples
Problem 1
Factorize mΒ²+ mn + nΒ²
We know that, 1+π+π2=0
⇒ π+π2= -1 ……….(1)
And, π3=1 …….(2)
So, m2+mn+n2
= m2 - (-1)mn +1× n2
= m2 - (π+π2)mn + π3× n2 [Using (1) and (2)]
= m2- mnπ- mnπ2+ n2π3
= m(m-nπ) -nπ2(m-nπ)
= (m-nπ)(m-nπ2)
Answer : (m-nπ)(m-nπ2)
Explanation
We used the properties of the cube root of unity to factorise the expression.
Problem 2
Find πβΆβΆ
π66
=(π3)22
=(1)22
=1
Answer: 1
Explanation
We used the property π3=1, and solved the expression.
Problem 3
Prove that (1+π)Β³+(1+πΒ²)Β³ = -2
We know that, 1+π+π2=0
⇒1+π= -π2 ……….(1)
And also, 1+π2= -π ………(2)
LHS = (1+π)3+(1+π2)3
=(-π2)3+(-π)3 [Using (1) and (2)]
=(-π6)+(-π3)
= -(π3)2 - (π3)
=-(1)2 - 1 [using the property π3=1]
= -1-1
=-2
=RHS [proved]
Explanation
We proved the given expression to be true using properties of cube root if unity like 1+π+π2=0 and π3=1.
Problem 4
Prove that (1+π-πΒ²)βΆ= -64
We know that, 1+π+π2=0
⇒1+ π= -π2 ……….(1)
And, π3=1 …….(2)
LHS
= (1+π-π2)6
=(-π2-π2)6 [using (1)]
=(-2π2)6
=26 × (-π2)6
=64× (-π12)
= 64× (-(π3)4)
= 64× (-(1)4)
= 64× (-1)
= -64
=RHS
Explanation
LHS=RHS
Hence proved
FAQs for Cube Root of Unity
1.What is the cube root of unity rule?
2.What is the expression of the cube root of unity ?
3.How do you find the cube root of unity?
4. Is the cube root of unity 1?
5.How does learning Algebra help students in Bahrain make better decisions in daily life?
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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.
- 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.
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Jaskaran Singh Saluja
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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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