Nth Root

N^th root of unity is the root of unity when taken which on taking to the power n gives the value 1. N^th 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 n^th 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 n^th root of any number as suppose we take n^th power of any number 'a'

aⁿ = b

then, the n^th root of 'b' is 'a' we represent this as,

ⁿ√b = a

We can also check for the n^th root of unity as,

zⁿ = 1

then, the n^th root of '1' is 'z' we represent this as,

ⁿ√1 = z

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

What is N^th Root?

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

We can mathematically express this as if the n^th 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

N^th Root Symbol

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

N^th root of Unity

N^th root of unity is the specific case of n^th root of any number we define n^th 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 n^th root of unity is always 1 but, if we consider complex number things gets more interesting when we consider the complex number "i", "ωおめが" and other roots of unity comes to play.

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

zⁿ = 1

gives the n^th root of unity.

We can easily solve this equation using complex numbers.

How to Find n^th Root of Unity?

N^th 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ぱい) + i sin (0+2mπぱい)                    (where m∈N)

Taking n^th root on both sides,

z = [cos (2mπぱい) + i sin (2mπぱい)]^1/n

Using DeMoivre's Theorem

z = [cos (2mπぱい/n) + i sin (2mπぱい/n)]

This can be represented in Euler Form,

z = e^{(i2mπぱい/n)}

This is the n^th root of unity for m ∈ N

N^th Root of Unity in Complex Numbers

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

Comparing the n^th root of unity to a general complex number we get,

x + iy = [cos (2mπぱい/n) + i sin (2mπぱい/n)]

⇒ x = cos (2mπぱい/n)...(i)

and y = sin (2mπぱい/n)...(ii)

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

x² + y² = cos²(2mπぱい/n) + sin²(2mπぱい/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おめが,

ωおめが =  e^{(i2mπぱい/n)}

Taking power n both sides,

(ωおめが)ⁿ =  e^{(i2mπぱい/n)n}

⇒ ωおめがⁿ =  e^i2mπぱい

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

1, ωおめが, ωおめが², ωおめが³,...ωおめが^n-1 

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

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Properties of N^th Root of Unity

• N^th 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おめが = (-1/2 + i√(3)/2), ωおめが² =(-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おめが + ωおめが² = 0

• All the n^th roots of unity are in GP i.e. 1, ωおめが², ωおめが³, ωおめが⁴, ...., ωおめが^n-1, are in GP
• The product of all the roots of unity is,

1 × ωおめが² × ωおめが³ × ωおめが⁴ × ....× ωおめが^n-1 = (-1)^n-1

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Solved Examples on N^th 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) = 2^4/4x^8/2

⇒  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