Summe.org Mathematical Calculators & Solvers

Tetration Calculator

Calculate tetration, power towers, and repeated exponentiation. Enter a base and tower height to calculate a ↑↑ h.

Tetration Calculator

Tetration is repeated exponentiation. For integer height, the operation creates a power tower of the selected base.

Example: 2, 3, 4, or 2.5
Use a non-negative integer height.

What Is Tetration?

Tetration is an operation obtained by repeatedly applying exponentiation. It can be viewed as exponentiation performed repeatedly to create a power tower.

Tetration notation:
a ↑↑ h

Here a is the base and h is the height of the tower.

For example:

3 ↑↑ 1 = 3

3 ↑↑ 2 = 3³ = 27

3 ↑↑ 3 = 3^(3³) = 3²⁷

The power tower is evaluated from the top downward. This right-associated structure is important because a^(a^a) is generally very different from (a^a)^a.

Tetration Examples

Small tetration values grow extremely quickly. The following examples illustrate how the height changes the power tower.

Tetration Power Tower Value
2 ↑↑ 1 2 2
2 ↑↑ 2 2² 4
2 ↑↑ 3 2^(2²) 16
2 ↑↑ 4 2^(2^(2²)) 65,536
3 ↑↑ 3 3^(3³) 7,625,597,484,987

Tetration vs. Exponentiation

Exponentiation repeats multiplication conceptually, while tetration repeats exponentiation.

Multiplication

Repeated addition leads to multiplication.

Exponentiation

Repeated multiplication produces powers such as an.

Tetration

Repeated exponentiation creates a power tower.

Tetration and Hyperoperations

Tetration belongs to the hierarchy of hyperoperations. In Knuth's up-arrow notation, tetration is represented using two up arrows:

a ↑↑ h

It follows exponentiation in the usual hyperoperation hierarchy. Higher operations include pentation and hexation.

Hyperoperation Calculator

Explore tetration and higher hyperoperations.

Pentation Calculator

Calculate the next major operation after tetration.

Hexation Calculator

Explore the higher hyperoperation after pentation.

Related Math Calculators

Continue exploring other mathematical operations and calculators on Summe.org.

Sigma Calculator

Evaluate finite summations using sigma notation.

Sum of a Series

Calculate arithmetic and geometric series sums.

Factorial Sum

Calculate sums involving factorial expressions.

Number Partition

Explore integer partitions of a number.

Derivative Calculator

Calculate derivatives of mathematical functions.

Digital Root Calculator

Calculate the digital root of an integer.

How the Tetration Calculator Works

If the height is a positive integer, the calculator builds a power tower containing the base repeatedly.

a ↑↑ 1 = a

a ↑↑ 2 = aa

a ↑↑ 3 = a(aa)

a ↑↑ 4 = a(a(aa))

Because tetration grows much faster than ordinary exponentiation, even relatively small bases and heights can produce numbers that are too large for ordinary computer number types.

When an exact decimal result becomes impractical, this calculator reports the scale of the result instead of attempting to display an unusably large integer.

Frequently Asked Questions

Tetration is repeated exponentiation. It creates a power tower in which the same base is used repeatedly.

The notation a ↑↑ h represents tetration with base a and height h. The two up arrows distinguish tetration from ordinary exponentiation.

A power tower is a stack of exponentiations. For example, 3 ↑↑ 4 represents 3^(3^(3^3)), evaluated from the top downward.

Tetration grows extraordinarily quickly. Increasing the height by only one level can make the result vastly larger than the previous result. Eventually the complete decimal expansion becomes impractical to store or display.

Yes. Tetration is commonly placed in the hyperoperation hierarchy after exponentiation. Pentation and hexation are higher operations in that hierarchy.