# Tetration

repeated or iterated exponentiation

Tetration is the hyperoperation which comes after exponentiation.[1] ${\displaystyle ^{x}{y}}$ means y exponentiated by itself, (x-1) times.[2][3][4] List of first 4 natural number hyperoperations, the inverse of tetration is the super root shown in the example

${\displaystyle a+n=a+\underbrace {1+1+\cdots +1} _{n}}$
n copies of 1 added to a.
2. Multiplication
${\displaystyle a\times n=\underbrace {a+a+\cdots +a} _{n}}$
n copies of a combined by addition.
3. Exponentiation
${\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}}$
n copies of a combined by multiplication.
4. Tetration
${\displaystyle {^{n}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} _{n}}$
n copies of a combined by exponentiation, right-to-left.

The above example is read as "the nth tetration of a".

## Examples

• ${\displaystyle ^{2}3=3^{3}=27}$
• ${\displaystyle ^{3}3=3^{({3^{3}})}=3^{27}=7,625,597,484,987}$
${\displaystyle x}$  ${\displaystyle {}^{2}x}$  ${\displaystyle {}^{3}x}$  ${\displaystyle {}^{4}x}$
1 1 (11) 1 (11) 1 (11)
2 4 (22) 16 (24) 65,536 (216)
3 27 (33) 7,625,597,484,987 (327) 1.258015 × 103,638,334,640,024
4 256 (44) 1.34078 ×10154 (4256) ${\displaystyle \exp _{10}^{3}(2.18726)}$  (8.1 × 10153 digits)
5 3,125 (55) 1.91101 × 102,184 (53,125) ${\displaystyle \exp _{10}^{3}(3.33928)}$  (1.3 × 102,184 digits)
6 46,656 (66) 2.65912 × 1036,305 (646,656) ${\displaystyle \exp _{10}^{3}(4.55997)}$  (2.1 × 1036,305 digits)
7 823,543 (77) 3.75982 × 10695,974 (7823,543) ${\displaystyle \exp _{10}^{3}(5.84259)}$  (3.2 × 10695,974 digits)
8 16,777,216 (88) 6.01452 × 1015,151,335 ${\displaystyle \exp _{10}^{3}(7.18045)}$  (5.4 × 1015,151,335 digits)
9 387,420,489 (99) 4.28125 × 10369,693,099 ${\displaystyle \exp _{10}^{3}(8.56784)}$  (4.1 × 10369,693,099 digits)
10 10,000,000,000 (1010) 1010,000,000,000 ${\displaystyle \exp _{10}^{4}(1)}$  (1010,000,000,000 digits)