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Last updated on November 30th, 2024

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Cube root of 1

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Foundation
Intermediate
Advance Topics

The cube root of a number is a number multiplied by itself thrice equals the original number. We apply cube roots in geometry for calculating volumes, to scale objects in physics and in engineering to measure density and many others. Let's learn more about the cube root of 1.

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What is the cube root of 1?

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


βˆ›1=1 


βˆ›1 has three roots→ 1,π›š, π›š2, which on multiplication together gives “1” as a product. 1×π›š×π›š2=1.

 

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

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Finding the Cube Root of 1

Now, let us find the meaning of π›š 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 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
 

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Important Glossaries for Cube root of 1

  • 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. 
     
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