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[change | change source]

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[change | change source]