Editing Tetration (section)

== Examples ==
Because of the extremely fast growth of tetration, most values in the following table are too large to write in [[scientific notation]]. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate. Usually, the limit that can be calculated in a numerical calculation program such as [[Wolfram Alpha]] is 3↑↑4, and the number of digits up to 3↑↑5 can be expressed.

{| class="wikitable"
|+Examples of tetration
!scope="col"| <math>x</math>
!scope="col"| <math>{}^{2}x</math>
!scope="col"| <math>{}^{3}x</math>
!scope="col"| <math>{}^{4}x</math>
!scope="col"| <math>{}^{5}x</math>
!scope="col"| <math>{}^{6}x</math>
!scope="col"| <math>{}^{7}x</math>
|- align=right
!scope="row"| 1
| 1
| 1
| 1
| 1
| 1
| 1
|- align=right
!scope="row"| 2
| 4 (2{{sup|2}})
| 16 (2{{sup|4}})
| 65,536 (2{{sup|16}})
| 2.00353 × 10{{sup|19,728}}
| <math>\exp_{10}^3(4.29508)</math> (10{{sup|6.03123×10{{sup|19,727}}}})
| <math>\exp_{10}^4(4.29508)</math>
|- align=right
!scope="row"| 3
| 27 (3{{sup|3}})
| 7,625,597,484,987 (3{{sup|27}})
| 1.25801 × 10{{sup|3,638,334,640,024}} <ref name="tdm">DiModica, Thomas. [https://github.com/TediusTimmy/CiteMyself/blob/trunk/Tetration/README.md Tetration Values.] Retrieved 15 October 2023.</ref>
| <math>\exp_{10}^4(1.09902)</math>
(10{{sup|6.00225×10{{sup|3,638,334,640,023}}}})
| <math>\exp_{10}^5(1.09902)</math>
| <math>\exp_{10}^6(1.09902)</math>
|- align=right
!scope="row"| 4
| 256 (4{{sup|4}})
| 1.34078 × 10{{sup|154}} (4{{sup|256}})
| <math>\exp_{10}^3(2.18726)</math> (10{{sup|8.0723×10{{sup|153}}}})
| <math>\exp_{10}^4(2.18726)</math>
| <math>\exp_{10}^5(2.18726)</math>
| <math>\exp_{10}^6(2.18726)</math>
|- align=right
!scope="row"| 5
| 3,125 (5{{sup|5}})
| 1.91101 × 10{{sup|2,184}} (5{{sup|3,125}})
| <math>\exp_{10}^3(3.33928)</math> (10{{sup|1.33574×10{{sup|2,184}}}})
| <math>\exp_{10}^4(3.33928)</math>
| <math>\exp_{10}^5(3.33928)</math>
| <math>\exp_{10}^6(3.33928)</math>
|- align=right
!scope="row"| 6
| 46,656 (6{{sup|6}})
| 2.65912 × 10{{sup|36,305}} (6{{sup|46,656}})
| <math>\exp_{10}^3(4.55997)</math> (10{{sup|2.0692×10{{sup|36,305}}}})
| <math>\exp_{10}^4(4.55997)</math>
| <math>\exp_{10}^5(4.55997)</math>
| <math>\exp_{10}^6(4.55997)</math>
|- align=right
!scope="row"| 7
| 823,543 (7{{sup|7}})
| 3.75982 × 10{{sup|695,974}} (7<sup>823,543</sup>)
| <math>\exp_{10}^3(5.84259)</math> (3.17742 × 10{{sup|695,974}} digits)
| <math>\exp_{10}^4(5.84259)</math>
| <math>\exp_{10}^5(5.84259)</math>
| <math>\exp_{10}^6(5.84259)</math>
|- align=right
!scope="row"| 8
| 16,777,216 (8{{sup|8}})
| 6.01452 × 10{{sup|15,151,335}}
| <math>\exp_{10}^3(7.18045)</math> (5.43165 × 10{{sup|15,151,335}} digits)
| <math>\exp_{10}^4(7.18045)</math>
| <math>\exp_{10}^5(7.18045)</math>
| <math>\exp_{10}^6(7.18045)</math>
|- align=right
!scope="row"| 9
| 387,420,489 (9{{sup|9}})
| 4.28125 × 10{{sup|369,693,099}}
| <math>\exp_{10}^3(8.56784)</math> (4.08535 × 10{{sup|369,693,099}} digits)
| <math>\exp_{10}^4(8.56784)</math>
| <math>\exp_{10}^5(8.56784)</math>
| <math>\exp_{10}^6(8.56784)</math>
|- align=right
!scope="row"| 10
| 10,000,000,000 (10{{sup|10}})
| 10{{sup|10,000,000,000}}
| <math>\exp_{10}^4(1)</math> (10{{sup|10,000,000,000}} + 1 digits)
| <math>\exp_{10}^5(1)</math>
| <math>\exp_{10}^6(1)</math>
| <math>\exp_{10}^7(1)</math>
|}
'''Remark:''' If {{mvar|x}} does not differ from 10 by orders of magnitude, then for all <math>k \ge3,~ ^mx =\exp_{10}^k z,~z>1 ~\Rightarrow~^{m+1}x = \exp_{10}^{k+1} z' \text{ with }z' \approx z</math>. For example, <math>z - z' < 1.5\cdot 10^{-15} \text{ for } x = 3 = k,~ m = 4</math> in the above table, and the difference is even smaller for the following rows.