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# Roots

An ${\displaystyle \scriptstyle n\,}$th complex root (root of degree ${\displaystyle \scriptstyle n\,}$) is one of the ${\displaystyle \scriptstyle n\,}$ complex solutions of

${\displaystyle z^{n}=a,\quad n\in \mathbb {N} ^{+},\,a,\,z\in \mathbb {C} .\,}$

## Real roots

An ${\displaystyle \scriptstyle n\,}$th real root (root of degree ${\displaystyle \scriptstyle n\,}$) is one of the real solutions of

${\displaystyle x^{n}=a,\quad n\in \mathbb {N} ^{+},\,a,\,x\in \mathbb {R} .\,}$

If ${\displaystyle \scriptstyle n\,}$ is an even positive integer, then the two real roots are

${\displaystyle x=\pm {\sqrt[{n}]{a}},\,}$

while if ${\displaystyle \scriptstyle n\,}$ is an odd positive integer, then the single real root is

${\displaystyle x={\sqrt[{n}]{a}},\,}$

where ${\displaystyle n}$ is the root index and a is the radicand.

### Surds

A surd is an algebraic irrational root, e.g. ${\displaystyle \scriptstyle {\sqrt[{3}]{2}}\,}$ is a cubic surd. The quadratic surd ${\displaystyle \scriptstyle {\sqrt[{2}]{27}}\,=\,3\,{\sqrt[{2}]{3}}\,}$ is a mixed surd (i.e. a rational number multiplied by a surd).

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

##### 1st iteration
• Addition:  S(S(⋯ "a times" ⋯ (S(n))))
, the sum n  +  a
, where  n
is the augend and  a
is the addend. (When addition is commutative both are simply called terms.)
• Subtraction:  P(P(⋯ "s times" ⋯ (P(n))))
, the difference n  −  s
, where  n
is the minuend and  s
is the subtrahend.
##### 2nd iteration
• Multiplication:  n + (n + (⋯ "k times" ⋯ (n + (n))))
, the product m  ⋅   k
, where  m
is the multiplicand and  k
is the multiplier.[3] (When multiplication is commutative both are simply called factors.)
• Division: the ratio n  /  d
, where  n
is the dividend and  d
is the divisor.
##### 3rd iteration
• Exponentiation (  d
as "degree",  b
as "base",  n
as "variable").
• Powers:  n  ⋅   (n  ⋅   (⋯ "d times" ⋯ (n  ⋅   (n))))
, written  n d
.
• Exponentials:  b  ⋅   (b  ⋅   (⋯ "n times" ⋯ (b  ⋅   (b))))
, written  b n
.
• Exponentiation inverses (  d
as "degree",  b
as "base",  n
as "variable").
##### 5th iteration
• Pentation (  d
as "degree",  b
as "base",  n
as "variable").
• Penta-powers:  n ^^ (n ^^ (⋯ "d times" ⋯ (n ^^ (n ^^ (n)))))
, written  n ^^^ d or n ↑↑↑ d
.
• Penta-exponentials:  b ^^ (b ^^ (⋯ "n times" ⋯ (b ^^ (b ^^ (b)))))
, written  b ^^^ n or b ↑↑↑ n
.
• Pentation inverses
##### 6th iteration
• Hexation (  d
as "degree",  b
as "base",  n
as "variable").
• Hexa-powers:  n ^^^ (n ^^^ (⋯ "d times" ⋯ (n ^^^ (n))))
, written  n ^^^^ d or n ↑↑↑↑ d
.
• Hexa-exponentials:  b ^^^ (b ^^^ (⋯ "n times" ⋯ (b ^^^ (b))))
, written  b ^^^^ n or b ↑↑↑↑ n
.
• Hexation inverses
##### 7th iteration
• Heptation (  d
as "degree",  b
as "base",  n
as "variable").
• Hepta-powers:  n ^^^^ (n ^^^^ (⋯ "d times" ⋯ (n ^^^^ (n))))
, written  n ^^^^^ d or n ↑↑↑↑↑ d
.
• Hepta-exponentials:  b ^^^^ (b ^^^^ (⋯ "n times" ⋯ (b ^^^^ (b))))
, written  b ^^^^^ n or b ↑↑↑↑↑ n
.
• Heptation inverses
##### 8th iteration
• Octation (  d
as "degree",  b
as "base",  n
as "variable").
• Octa-powers:  n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n))))
, written  n ^^^^^^ d or n ↑↑↑↑↑↑ d
.
• Octa-exponentials:  b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b))))
, written  b ^^^^^^ n or b ↑↑↑↑↑↑ n
.
• Octation inverses

## Notes

1. 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
.