Tetration

Tetration is the hyperoperation which comes after exponentiation. [math]\displaystyle{ ^{x}{y} }[/math] means y exponentiated by itself, (x-1) times. List of first 4 natural number hyperoperations:

1. Addition

   [math]\displaystyle{ a + n = a\!\underbrace{''{}^{\cdots}{}'}_n }[/math]

   a succeeded n times.

2. Multiplication

   [math]\displaystyle{ a \times n = \underbrace{a + a + \cdots + a}_n }[/math]

   a added to itself, n times.

3. Exponentiation

   [math]\displaystyle{ a^n = \underbrace{a \times a \times \cdots \times a}_n }[/math]

   a multiplied by itself, n times.

4. Tetration

   [math]\displaystyle{ {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n }[/math]

   Note (operator associativity): [math]\displaystyle{ {^{n}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_n }[/math]

   a exponentiated by itself, n-1 times.

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

Example

For the example, addition is assumed.

1. [math]\displaystyle{ {^{2}3} = }[/math]

   [math]\displaystyle{ {3^{3}} = }[/math]

   [math]\displaystyle{ {3 \times 3 \times 3} = }[/math]

   [math]\displaystyle{ {3 \times (3 + 3 + 3)} = }[/math]

   [math]\displaystyle{ {3 \times {9}} = }[/math]

   [math]\displaystyle{ {3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 9 + 9 + 9} = }[/math]

   [math]\displaystyle{ 27 }[/math]

References

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