Tetration

Tetration is the hyperoperation which comes after exponentiation. <math>^{x}{y}</math> 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

   <math>a + n = a + \underbrace{1 + 1 + \cdots + 1}_n</math>

   n copies of 1 added to a.

2. Multiplication

   <math>a \times n = \underbrace{a + a + \cdots + a}_n</math>

   n copies of a combined by addition.

3. Exponentiation

   <math>a^n = \underbrace{a \times a \times \cdots \times a}_n</math>

   n copies of a combined by multiplication.

4. Tetration

   <math>{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math>

n copies of a combined by exponentiation, right-to-left.

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

Examples

• <math> ^{2}3 = 3^3 = 27 </math>
• <math> ^{3}3 = 3^{({3^3})} = 3^{27} = 7,625,597,484,987 </math>

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