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:

1. Addition

   $a + n = a\!\underbrace{''{}^{\cdots}{}'}_n$

   a succeeded n times.

2. Multiplication

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

   a added to itself, n times.

3. Exponentiation

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

   a multiplied by itself, n times.

4. Tetration

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

   Note (operator associativity): ${^{n}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_n$

   a exponentiated by itself, n-1 times.

Example

For the example, addition is assumed.

1. ${^{2}3} =$

   ${3^{3}} =$

   ${3 \times 3 \times 3} =$

   ${3 \times (3 + 3 + 3)} =$

   ${3 \times {9}} =$

   ${3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 9 + 9 + 9} =$

   $27$