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

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
![{\displaystyle x=\pm {\sqrt[{n}]{a}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/cbd30655216ed3cd22aaed5be5774351c48622c9)
while if
is an odd positive integer, then the single real root is
![{\displaystyle x={\sqrt[{n}]{a}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7afa8573fe7f7bf49ac9cae28efae0b2aabe4942)
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]
0th iteration
1st iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2nd iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.[3] (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3rd iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes
Notes