Cube root of 1

∛1 — is the symbolic representation of ‘cube root of 1’. 

∛1=1 

∛1 has three roots→ 1,𝛚, 𝛚², which on multiplication together gives “1” as a product. 1×𝛚×𝛚²=1.

As mentioned above, the cube root of 1 or 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 1 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.
 

Now, let us find the meaning of 𝛚 here. To find the cube root of 1, we will make use of some algebraic formulas. We know that, the cube root of 1 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
 

• Natural numbers — Numbers that range from 1 to infinity. 

• Whole numbers — The whole numbers are part of the number system, which includes all the positive integers from 0 to infinity. 

• Integers — Integers can be a positive natural number, negative of a positive number, or zero. We can perform all the arithmetic operations on integers. The examples of integers are, 1, 2, 5,8, -8, -12, etc.

• Square root  — The square root of a number is a value “y” such that when “y” is multiplied by itself → y × y, the result is the original number.