nth root

nth root

The phrase “

the

n

-th root of a number

” is a somewhat misleading

concept

that requires a fair amount of thought to make

rigorous

.

For

n

a

positive integer

, we define

an

n

-th root of a number

x

to be a number

y

such that

y

n

=

x

. The number

n

is said to be the

index

of the root. Note that the term “number” here is ambiguous, as the discussion can apply in a

variety

of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with

n

-th roots of real and

complex numbers

.

In an effort to give meaning to the term

the

n

-th root of a

real number

x

, we define it to be the unique real number that

y

is

an

n

th root of

x

and such that

sign

⁡

(

x

)

=

sign

⁡

(

y

)

, if such a number exists. We denote this number by

x

n

, or by

x

1

n

if

x

is

positive

. This specific

n

th root is also called the

principal

n

th root

.

Example:

81

4

=

3

because

3

4

=

3

×

3

×

3

×

3

=

81

, and

3

is the unique positive real number with this

property

.

Example: If

x

+

1

is a positive real number, then we can write

x

5

+

5

⁢

x

4

+

10

⁢

x

3

+

10

⁢

x

2

+

5

⁢

x

+

1

5

=

x

+

1

because

(

x

+

1

)

5

=

(

x

2

+

2

⁢

x

+

1

)

2

⁢

(

x

+

1

)

=

x

5

+

5

⁢

x

4

+

10

⁢

x

3

+

10

⁢

x

2

+

5

⁢

x

+

1

. (See the

Binomial Theorem

and .)

The

nth root

operation

is

distributive

for

multiplication

and

division

, but not for

addition

and

subtraction

. That is,

x

×

y

n

=

x

n

×

y

n

, and

x

y

n

=

x

n

y

n

. However, except in special cases,

x

+

y

n

≠

x

n

+

y

n

and

x

-

y

n

≠

x

n

-

y

n

.

Example:

81

625

4

=

3

5

because

(

3

5

)

4

=

3

4

5

4

=

81

625

.

Note that when we restrict our attention to real numbers,

expressions

like

-

3

are undefined. Thus, for a more full

definition

of

n

th roots, we will have to incorporate the notion of complex numbers:

The nth roots of a complex number

t

=

x

+

y

⁢

i

are all the complex numbers

z

1

,

z

2

,

…

,

z

n

∈

ℂ

that

satisfy

the condition

z

k

n

=

t

. Applying the

fundamental theorem of algebra

(

complex

version) to the

function

x

n

-

t

tells us that

n

such complex numbers always exist (counting

multiplicity

).

One of the more popular methods of finding these roots is through

trigonometry

and the

geometry

of complex numbers. For a complex number

z

=

x

+

i

⁢

y

, recall that we can put

z

in polar form:

z

=

(

r

,

θ

)

, where

r

=

x

2

+

y

2

2

, and

θ

=

π

2

if

x

=

0

, and

θ

=

arctan

⁡

y

x

if

x

≠

0

. (See the

Pythagorean Theorem

.) For the specific procedures involved, see calculating the nth roots of a complex number.

Title

nth root

Canonical

name

NthRoot

Date of creation

2013-03-22 11:57:27

Last modified on

2013-03-22 11:57:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

25

Author

mathcam (2727)

Entry type

Definition

Classification

msc 30-00

Classification

msc 12D99

Synonym

complex root

Synonym

principal root

Related topic

SquareRoot

Related topic

CubeRoot

Related topic

RealNumber

Related topic

RationalNumber

Related topic

Complex

Related topic

IrrationalNumber

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EvenEvenOddRule

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ExtensionOfValuationFromCompleteBaseField

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Radical5

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Radical6

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ExampleOfConvergingIncreasingSequence

Related topic

NthRootByNewtonsMethod

Defines

index

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