# Nth Root

Last Updated : 18 Apr, 2024

Nth root of unity is the root of unity when taken which on taking to the power n gives the value 1. Nth root of any number is defined as the number that takes to the power of n results in the original number. For example, if we take the nth root of any number, say b, the result is a, and then a is raised to n power and will get b.

We define the nth root of any number as suppose we take nth power of any number ‘a’

an = b

then, the nth root of ‘b’ is ‘a’ we represent this as,

nâˆšb = a

We can also check for the nth root of unity as,

zn = 1

then, the nth root of ‘1 is ‘z we represent this as,

nâˆš1 = z

In this article, we will learn about, the nth root of any number, the nth root of unity, and others in detail.

## What is Nth Root?

We define the nth root of any number as the number which when taken to nth power results in the original number, on multiplying the nth of any number n times we get the original number.

We can mathematically express this as if the nth root of x is y then,

nâˆšx = y

â‡’ yn = x

This can also be represented as,

nâˆšx Ã— nâˆšx Ã— nâˆšx Ã— nâˆšx …. n times = x

Example: Find the third root of 8

Solution:

We know that,

23 = 2Ã—2Ã—2 = 8

3âˆš8 = Â 3âˆš(2Ã—2Ã—2) = 2

Thus, the third root of 8 is 2

## Nth Root Symbol

The nth root of any number is represented using the symbol, nâˆšx, here, we find the nth root of x.Â

In the expression nâˆšx, x is called the radicand of the term and n is the index of the term. We can also represent this as the exponent of x in fraction, i.e.

nâˆšx = (x)1/n

## Nth root of Unity

Nth root of unity is the specific case of nth root of any number we define nth root of unity as the number which when multiplying n times gives n. As we know for real numbers if any number is multiplied by 1 we get, the number itself, and 1 when multiplied by itself a finite number of times results in 1. Thus, in the case of real numbers the nth root of unity is always 1 but, if we consider complex number things gets more interesting when we consider the complex number “i”, “Ï‰” and other roots of unity comes to play.

Thus, we have various roots of unity. The solution to the equation,

zn = 1

gives the nth root of unity.

We can easily solve this equation using complex numbers.

## How to Find nth Root of Unity?

Nth root of unity can be easily found by finding the solution to the equation,

zn = 1

In polar form, we write this equation as,

zn = cos 0 + i sin 0

In general form,

zn = cos (0+2mÏ€) + i sin (0+2mÏ€) Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (where mâˆˆN)

Taking nth root on both sides,

z = [cos (2mÏ€) + i sin (2mÏ€)]1/n

Using DeMoivre’s Theorem

z = [cos (2mÏ€/n) + i sin (2mÏ€/n)]

This can be represented in Euler Form,

z = e(i2mÏ€/n)

This is the nth root of unity for m âˆˆ N

## Nth Root of Unity in Complex Numbers

We know that the general form of a complex number is x + iy.

Comparing the nth root of unity to a general complex number we get,

x + iy = [cos (2mÏ€/n) + i sin (2mÏ€/n)]

â‡’ x = cos (2mÏ€/n)…(i)

and y = sin (2mÏ€/n)…(ii)

Squaring and adding (i) and (ii) we get,

x2 + y2 = cos2(2mÏ€/n) + sin2(2mÏ€/n) = 1

The above equation is the equation of the circle with the centre at origin (0,0) and radius 1.

If we represent the complex number as Ï‰,

Ï‰ = Â e(i2mÏ€/n)

Taking power n both sides,

(Ï‰)n = Â e(i2mÏ€/n)n

â‡’ Ï‰n = Â ei2mÏ€

This gives the nth root of unity taking n â‰¥ 0, we get the root of unity as,

1, Ï‰, Ï‰2, Ï‰3,…Ï‰n-1Â

These roots can be represented in a unit circle in a complex plane as,

## Properties of Nth Root of Unity

• Nth root of unity is the points on the circumference of the circle in the complex plane with the center on the origin and a radius of 1 unit.
• 1 and -1 are the 2nd roots of unity.
• 1, -1. i, and -1 are the 4th roots of unity.
• i = âˆš(-1) and i2 = -1, i4 = 1
• 1, Ï‰ = (-1/2 + iâˆš(3)/2), Ï‰2 =(-1/2 – iâˆš(3)/2) are the three roots of the unity.
• Multiplying two imaginary cube roots of unity results in 1.
• The sum of all three cube roots of unity, i.e.

1 + Ï‰ + Ï‰2 = 0

• All the nth roots of unity are in GP i.e. 1, Ï‰2, Ï‰3, Ï‰4, …., Ï‰n-1, are in GP
• The product of all the roots of unity is,

1 Ã— Ï‰2 Ã— Ï‰3 Ã— Ï‰4 Ã— ….Ã— Ï‰n-1 = (-1)n-1

## Solved Examples on Nth Root

Example 1: Find 5âˆš(-243)

Solution:

As we know,

(-3)5 = -243

Thus,5âˆš(-243) = Â (-3)5Â

and (-3)5 =[ (-3)5]1/5 Â = -3

Thus, 5âˆš(-243) = -3.

Example 2: Simplify 4âˆš(16x8)

Solution:

As 16 = 2Ã—2Ã—2Ã—2 = 24

Thus, 4âˆš(16x8) = 4âˆš(24x8)

â‡’ Â 4âˆš(16x8) = 24/4x8/2

â‡’ Â 4âˆš(16x8) = 2x2

Example 3: Simplify 4âˆš(-x24)

Solution:

4âˆš(-x24) = 4âˆš[(-1)Ã—(x24)]

â‡’ 4âˆš(-x24) = (-1)1/4 Ã—(x24)1/4

â‡’ 4âˆš(-x24) = ix8

## FAQs on Nth Root

### What is the nth root?

The nth root of unity is the number which when multiplied n times gives the original number. If the nth root of unity is x1/n = b, then

bn = 1

### What is a Square Root?

The square root of the number is the number which when multiplied by itself gives the original number. It is represented by the symbol âˆš, Now the square root of (x) is represented as, âˆš(x). For example, the square root of 4 is,

âˆš(4) =âˆš(2Ã—2) = 2

### What is a Cube Root?

The cube root of the number is the number which when multiplied by itself thrice gives the original number. It is represented by the symbol 3âˆš, Now the cube root of (x) is represented as, 3âˆš(x). For example, the cube root of 8 is,

âˆš(8) = âˆš(2Ã—2Ã—2) = 2

### What are the Cube Roots of Unity?

Cube root of unity is the number which when multiplied with itself three times gives results as 1. We can represent this as,

3âˆš(1) = 1, Ï‰, Ï‰2

Here, 1, Ï‰, and Ï‰2 are the cube root of unity.

### What is the fourth root of unity?

Fourth root of unity is the number which when multiplied with itself four times gives results as 1. We can represent this as,

4âˆš(1) = 1, -1, i, -i

Here, 1, -1, i, and -i are the fourth roots of the unity.

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