Tetration

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 operations of tetration are the super root and the super logarithm.

1. Addition

   ${\displaystyle a+n=a+\underbrace {1+1+\cdots +1} _{n}}$

   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".

• ${\displaystyle ^{2}3=3^{3}=27}$
• ${\displaystyle ^{3}3=3^{({3^{3}})}=3^{27}=7,625,597,484,987}$

${\displaystyle x}$	${\displaystyle {}^{2}x}$	${\displaystyle {}^{3}x}$	${\displaystyle {}^{4}x}$	${\displaystyle {}^{\infty }x}$
1	1 (11)	1 (11)	1 (11)	${\displaystyle {}^{\infty }1=1}$ (11) (Absolute infinity and Infinity)
2	4 (22)	16 (24)	65,536 (216)	${\displaystyle \exp _{10}^{\infty -3}(4.29508)}$
3	27 (33)	7,625,597,484,987 (327)	1.258015 × 103,638,334,640,024	${\displaystyle \exp _{10}^{\infty -1}(1.09902)}$
4	256 (44)	1.34078 ×10154 (4256)	${\displaystyle \exp _{10}^{3}(2.18726)}$ (8.1 × 10153 digits)	${\displaystyle \exp _{10}^{\infty -1}(2.18726)}$
5	3,125 (55)	1.91101 × 102,184 (53,125)	${\displaystyle \exp _{10}^{3}(3.33928)}$ (1.3 × 102,184 digits)	${\displaystyle \exp _{10}^{\infty -1}(3.33928)}$
6	46,656 (66)	2.65912 × 1036,305 (646,656)	${\displaystyle \exp _{10}^{3}(4.55997)}$ (2.1 × 1036,305 digits)	${\displaystyle \exp _{10}^{\infty -1}(4.55997)}$
7	823,543 (77)	3.75982 × 10695,974 (7823,543)	${\displaystyle \exp _{10}^{3}(5.84259)}$ (3.2 × 10695,974 digits)	${\displaystyle \exp _{10}^{\infty -1}(5.84259)}$
8	16,777,216 (88)	6.01452 × 1015,151,335	${\displaystyle \exp _{10}^{3}(7.18045)}$ (5.4 × 1015,151,335 digits)	${\displaystyle \exp _{10}^{\infty -1}(7.18045)}$
9	387,420,489 (99)	4.28125 × 10369,693,099	${\displaystyle \exp _{10}^{3}(8.56784)}$ (4.1 × 10369,693,099 digits)	${\displaystyle \exp _{10}^{\infty -1}(8.56784)}$
10	10,000,000,000 (1010)	1010,000,000,000	${\displaystyle \exp _{10}^{4}(1)}$ (1010,000,000,000 digits)	${\displaystyle \exp _{10}^{\infty }(1)}$
∞	∞∞	${\displaystyle \exp _{10}^{3}(\infty )}$ (∞∞∞)	${\displaystyle \exp _{10}^{4}(\infty )}$	${\displaystyle \exp _{10}^{\infty }(\infty )}$