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# Roots

(Redirected from Root extraction)

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

$z^n = a,\quad n \in \N^+,\, a,\, z \in \C. \,$

## Real roots

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

$x^n = a,\quad n \in \N^+,\, a,\, x \in \R. \,$

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

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

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

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

### Surds

A surd is an algebraic irrational root, e.g. $\scriptstyle \sqrt[3]{2} \,$ is a cubic surd. The quadratic surd $\scriptstyle \sqrt[2]{27} \,=\, 3 \, \sqrt[2]{3} \,$ 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(... s times ...(S(n)))), the sum n+s
• Subtraction, P(P(... s times ...(P(n)))), the difference n-s
##### 5th iteration
• Pentation (d as "dimension", 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 "dimension", 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 "dimension", 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 "dimension", 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
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