Tetration

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

1. Addition

   $a+n=a+\underset{n}{\underbrace{1+1+\cdots +1}}$

   Template:Mvar copies of 1 added to Template:Mvar.

2. Multiplication

   $a\times n=\underset{n}{\underbrace{a+a+\cdots +a}}$

   Template:Mvar copies of Template:Mvar combined by addition.

3. Exponentiation

   $a^n=\underset{n}{\underbrace{a\times a\times \cdots \times a}}$

   Template:Mvar copies of Template:Mvar combined by multiplication.

4. Tetration

   ${}^{n}a=\underset{n}{\underbrace{a^{a^{{\cdot }^{{\cdot }^a}}}}}$

Template:Mvar copies of Template:Mvar combined by exponentiation, right-to-left.

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

Examples

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

$x$	${}^{2}x$	${}^{3}x$	${}^{4}x$
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)	${\exp }_{10}^3(2.18726)$ (8.1 × 10153 digits)
5	3,125 (55)	1.91101 × 102,184 (53,125)	${\exp }_{10}^3(3.33928)$ (1.3 × 102,184 digits)
6	46,656 (66)	2.65912 × 1036,305 (646,656)	${\exp }_{10}^3(4.55997)$ (2.1 × 1036,305 digits)
7	823,543 (77)	3.75982 × 10695,974 (7823,543)	${\exp }_{10}^3(5.84259)$ (3.2 × 10695,974 digits)
8	16,777,216 (88)	6.01452 × 1015,151,335	${\exp }_{10}^3(7.18045)$ (5.4 × 1015,151,335 digits)
9	387,420,489 (99)	4.28125 × 10369,693,099	${\exp }_{10}^3(8.56784)$ (4.1 × 10369,693,099 digits)
10	10,000,000,000 (1010)	1010,000,000,000	${\exp }_{10}^4(1)$ (1010,000,000,000 digits)

References

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