This site is supported by donations to The OEIS Foundation.

# Roots

From OeisWiki

(Redirected from Root extraction)

An th **complex root** (**root** of degree ) is one of the complex solutions of

## Contents

## Real roots

An th **real root** (**root** of degree ) is one of the real solutions of

If is an even positive integer, then the two real roots are

while if is an odd positive integer, then the single real root is

where is the root **index** and a is the **radicand**.

### Surds

A **surd** is an algebraic irrational root, e.g. is a cubic surd. The quadratic surd is a **mixed surd** (i.e. a rational number multiplied by a surd).

## See also

#### Hierarchical list of operations pertaining to numbers ^{[1]} ^{[2]}

##### 0^{th} iteration

- Successor:

.S( *n*) - Predecessor:

.P( *n*)

##### 1^{st} iteration

- Addition:

, theS(S(⋯ " *a*times" ⋯ (S(*n*))))*sum*

, where*n*+*a*

is the*n**augend*and

is the*a**addend*. (When addition is commutative both are simply called*terms*.) - Subtraction:

, theP(P(⋯ " *s*times" ⋯ (P(*n*))))*difference*

, where*n*−*s*

is the*n**minuend*and

is the*s**subtrahend*.

##### 2^{nd} iteration

- Multiplication:

, the*n*+ (*n*+ (⋯ "*k*times" ⋯ (*n*+ (*n*))))*product*

, where*m*⋅*k*

is the*m**multiplicand*and

is the*k**multiplier*.^{[3]}(When multiplication is commutative both are simply called*factors*.) - Division: the
*ratio*

, where*n*/*d*

is the*n**dividend*and

is the*d**divisor*.- Quotient: (integer division).
- Remainder: (modulo and congruences).

##### 3^{rd} iteration

- Exponentiation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Powers:

, written*n*⋅ (*n*⋅ (⋯ "*d*times" ⋯ (*n*⋅ (*n*))))

.*n**d* - Exponentials:

, written*b*⋅ (*b*⋅ (⋯ "*n*times" ⋯ (*b*⋅ (*b*))))

.*b**n*- Exponential function:

, where*e**n*

is Euler's number.*e*

- Exponential function:

- Powers:
- Exponentiation inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Roots:

.*d*√*n* - Logarithms:

.log *b**n*- Natural logarithm function:

, orlog *n*

, wherelog *e**n*

is Euler's number.*e*

- Natural logarithm function:

- Roots:

##### 4^{th} iteration

- Tetration (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-powers (super-powers):

, written*n*^ (*n*^ (⋯ "*d*times" ⋯ (*n*^ (*n*))))

.*n*^^*d*or*n*↑↑*d* - Tetra-exponentials (super-exponentials):

, written*b*^ (*b*^ (⋯ "*n*times" ⋯ (*b*^ (*b*))))

.*b*^^*n*or*b*↑↑*n*

- Tetra-powers (super-powers):
- Tetration inverses (

as "degree",*d*

as "base",*b*

as "variable").*n*- Tetra-roots (super-roots)
- Tetra-logarithms (super-logarithms):

.slog *b**n*- Iterated logarithm:

.log ⁎ *b**n*= ⌈slog*b**n*⌉

- Iterated logarithm:

##### 5^{th} iteration

- Pentation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Penta-powers:

, written*n*^^ (*n*^^ (⋯ "*d*times" ⋯ (*n*^^ (*n*^^ (*n*)))))

.*n*^^^*d*or*n*↑↑↑*d* - Penta-exponentials:

, written*b*^^ (*b*^^ (⋯ "*n*times" ⋯ (*b*^^ (*b*^^ (*b*)))))

.*b*^^^*n*or*b*↑↑↑*n*

- Penta-powers:
- Pentation inverses

##### 6^{th} iteration

- Hexation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hexa-powers:

, written*n*^^^ (*n*^^^ (⋯ "*d*times" ⋯ (*n*^^^ (*n*))))

.*n*^^^^*d*or*n*↑↑↑↑*d* - Hexa-exponentials:

, written*b*^^^ (*b*^^^ (⋯ "*n*times" ⋯ (*b*^^^ (*b*))))

.*b*^^^^*n*or*b*↑↑↑↑*n*

- Hexa-powers:
- Hexation inverses

##### 7^{th} iteration

- Heptation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Hepta-powers:

, written*n*^^^^ (*n*^^^^ (⋯ "*d*times" ⋯ (*n*^^^^ (*n*))))

.*n*^^^^^*d*or*n*↑↑↑↑↑*d* - Hepta-exponentials:

, written*b*^^^^ (*b*^^^^ (⋯ "*n*times" ⋯ (*b*^^^^ (*b*))))

.*b*^^^^^*n*or*b*↑↑↑↑↑*n*

- Hepta-powers:
- Heptation inverses

##### 8^{th} iteration

- Octation (

as "degree",*d*

as "base",*b*

as "variable").*n*- Octa-powers:

, written*n*^^^^^ (*n*^^^^^ (⋯ "*d*times" ⋯ (*n*^^^^^ (*n*))))

.*n*^^^^^^*d*or*n*↑↑↑↑↑↑*d* - Octa-exponentials:

, written*b*^^^^^ (*b*^^^^^ (⋯ "*n*times" ⋯ (*b*^^^^^ (*b*))))

.*b*^^^^^^*n*or*b*↑↑↑↑↑↑*n*

- Octa-powers:
- Octation inverses

## Notes

- ↑ Hyperoperation—Wikipedia.org.
- ↑ Grzegorczyk hierarchy—Wikipedia.org.
- ↑ There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:

.*ω*× 2**:**=*ω*+*ω*