Tetration — Definition, Formula & Examples

Tetration is repeated exponentiation. You stack copies of the same number into a tower of exponents, so

{}^{3}2

means

2^{2^{2}}

For a real base $a$ and a positive integer height $n$ , tetration is defined recursively by ${}^{1}a = a$ and ${}^{n+1}a = a^{\left({}^{n}a\right)}$ . By convention ${}^{0}a = 1$ . It is the fourth hyperoperation, after addition, multiplication, and exponentiation. In Knuth's up-arrow notation it is written $a \uparrow\uparrow n$ . The tower is right-associative: you evaluate it from the top exponent downward.

Key Formula

{}^{n}a = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \text{ copies of } a}, \qquad {}^{1}a = a, \quad {}^{n+1}a = a^{\left({}^{n}a\right)}

Where:

$a$ = the base, the number being stacked
$n$ = the height, the number of copies of a in the tower (a positive integer)
${}^{n}a$ = the tetration of a to height n, read "a tetrated to n"

Worked Example

Problem: Evaluate

{}^{4}2

, a power tower of four 2s.

Step 1: Start at the top of the tower. A tower of two 2s is an ordinary power.

{}^{2}2 = 2^{2} = 4

Step 2: Use the recursive rule. The tower of three 2s puts the previous result in the exponent.

{}^{3}2 = 2^{\left({}^{2}2\right)} = 2^{4} = 16

Step 3: Repeat once more for height 4.

{}^{4}2 = 2^{\left({}^{3}2\right)} = 2^{16} = 65536

Answer:

{}^{4}2 = 65536

. The next term,

{}^{5}2 = 2^{65536}

, already has 19,729 digits.

Why It Matters

Tetration shows how fast the hyperoperation sequence grows. That idea appears in computer science courses that study the Ackermann function and growth rates of algorithms. Up-arrow notation, built on tetration, is the standard way to describe enormous values such as Graham's number. The infinite tower

x^{x^{x^{\cdots}}}

is also a classic analysis problem: it converges exactly when

e^{-e} \le x \le e^{1/e}

Common Mistakes

Mistake: Evaluating the tower from the bottom up. For example, computing

{}^{3}3

(3^3)^3 = 27^3 = 19683

Correction: Power towers are right-associative, so you start with the top exponent:

{}^{3}3 = 3^{(3^3)} = 3^{27} = 7625597484987

. The bottom-up version simplifies to

3^{9}

and is far smaller.

Related Terms

Exponentiation — The operation tetration repeats; the previous hyperoperation
Exponent — Each level of the tower is an exponent
Power — Every intermediate tower value is a power
Product — Multiplication is the second hyperoperation
Arithmetic — Addition starts the hyperoperation sequence
Formula — Tetration is defined by a recursive formula