# Tetration

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Tetration is the hyperoperation which comes after exponentiation.[1] $^{x}{y}$ means y exponentiated by itself, (x-1) times.[2][3] 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. ${^{2}3} =$
${3^{3}} =$
${3 \times 3 \times 3} =$
${3 \times (3 + 3 + 3)} =$
${3 \times {9}} =$
${3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 9 + 9 + 9} =$
$27$

## References

1. "Google Answers: addition, multiplication, exponentiation, then ???". Answers.google.com. Retrieved 2011-11-02.
2. Daniel Geisler. "tetration.org". Tetration. Retrieved 2011-11-02.
3. "Power Tower - from Wolfram MathWorld". Mathworld.wolfram.com. Retrieved 2011-11-02.