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nth Roots

Last Updated : 08 Jun, 2023
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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 which on taking to the power of n results in the original number. For example, if we take nth root of any number say a and the result is b, and then b is raised to n power will get a.

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

an = b

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

n√b = a

We can also check for nth root of unity as,

zn = 1

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

n√1 = z

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

What is Nth Root?

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

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

n√x = y

⇒ yn = x

This can also be represented as,

n√x × n√x × n√x × n√x …. n times = x

Example: Find the third root of 8

Solution:

We know that,

23 = 2×2×2 = 8

3√8 =  3√(2×2×2) = 2

Thus, the third root of 8 is 2

Nth Root Symbol

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

In the expression n√x, 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.

n√x = (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”, “ω” 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+2mπ) + i sin (0+2mπ)                    (where m∈N)

Taking nth root on both sides,

z = [cos (2mπ) + i sin (2mπ)]1/n

Using DeMoivre’s Theorem

z = [cos (2mπ/n) + i sin (2mπ/n)]

This can be represented in Euler Form,

z = e(i2mπ/n)

This is the nth root of unity for m ∈ 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 (2mπ/n) + i sin (2mπ/n)]

⇒ x = cos (2mπ/n)…(i)

and y = sin (2mπ/n)…(ii)

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

x2 + y2 = cos2(2mπ/n) + sin2(2mπ/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 ω,

ω =  e(i2mπ/n)

Taking power n both sides,

(ω)n =  e(i2mπ/n)n

⇒ ωn =  ei2mπ

This gives the nth root of unity taking n ≥ 0, we get the root of unity as,

1, ω, ω2, ω3,…ωn-1 

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

Unit Circle

 

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 = √(-1) and i2 = -1, i4 = 1
  • 1, ω = (-1/2 + i√(3)/2), ω2 =(-1/2 – i√(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 + ω + ω2 = 0

  • All the nth roots of unity are in GP i.e. 1, ω2, ω3, ω4, …., ωn-1, are in GP
  • The product of all the roots of unity is,

1 × ω2 × ω3 × ω4 × ….× ωn-1 = (-1)n-1

Apart from this they also follow the Basic laws of exponents.

Read More,

Solved Examples on Nth Root

Example 1: Find 5√(-243)

Solution:

As we know,

(-3)5 = -243

Thus,5√(-243) =  (-3)5 

and (-3)5 =[ (-3)5]1/5  = -3

Thus, 5√(-243) = -3.

Example 2: Simplify 4√(16x8)

Solution:

As 16 = 2×2×2×2 = 24

Thus, 4√(16x8) = 4√(24x8)

⇒  4√(16x8) = 24/4x8/2

⇒  4√(16x8) = 2x2

Example 3: Simplify 4√(-x24)

Solution:

4√(-x24) = 4√[(-1)×(x24)]

4√(-x24) = (-1)1/4 ×(x24)1/4

4√(-x24) = ix8

FAQs on Nth Root

Q1: What is the nth root?

Answer:

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

Q2: What is a Square Root?

Answer:

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

√(4) =√(2×2) = 2

Q3: What is a Cube Root?

Answer:

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 3√, Now the cube root of (x) is represented as, 3√(x). For example, the cube root of 8 is,

√(8) = √(2×2×2) = 2

Q4: What are the Cube Roots of Unity?

Answer:

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

3√(1) = 1, ω, ω2

Here, 1, ω, and ω2 are the cube root of unity.

Q5: What is the fourth root of unity?

Answer:

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

4√(1) = 1, -1, i, -i

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



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