Calculate with Tetration
Your result will appear here.
What is a tetration calculator?
Tetration is iterated exponentiation: a repeated-base power tower evaluated from the top exponent down. For example, 2↑↑3 means 2^(2^2), not (2^2)^2. This calculator is limited to integer bases from 0 through 10 and heights from 0 through 5; towers outside its scientific-notation range are rejected rather than silently overflowing.
Important terms
- Tetration
- Repeated exponentiation, which builds a tower of equal bases.
- Base
- The integer repeated at each level of the tower.
- Height
- The number of copies of the base in the tower.
- Right-associative
- The topmost exponent is evaluated first, so a^(a^a) is the order for a height-three tower.
When this method is useful
- Exploring how quickly repeated exponentiation grows.
- Learning double-arrow notation for power towers.
- Comparing exponentiation with the next hyperoperation.
- Checking small finite towers without manually evaluating from the top.
Tetration Calculator
Power towers are evaluated from the top down
Formula and variables
- a
- the integer base from 0 to 10
- h
- the tower height from 0 to 5
- T(a,h)
- the value of a power tower with h copies of a
The recurrence starts with the empty-tower convention T(a,0)=1. Each new level raises a to the value of the complete tower one level shorter, making the expression right-associated.
Domain note: Integer bases 0–10 and heights 0–5 are supported. A zero base at height 2 or more reaches 0⁰ and is undefined. Some allowed input pairs exceed the supported numeric range and are rejected. Real or complex bases, fractional heights, and infinite towers are not supported.
How the formula works
How to use this calculator
- 1
Choose an integer base from 0 through 10.
- 2
Choose a tower height from 0 through 5.
- 3
Read the repeated-base tower; height counts copies of the base, not exponent signs.
- 4
Evaluate from the top down using T(a,h)=a^T(a,h−1).
- 5
Check whether the result is exact or rounded scientific notation.
Step-by-step examples
Example 1
Problem: Evaluate 2↑↑3.
Substitution: 2^(2^2) = 2^4
Result: 16
The height-three tower contains three copies of 2. Evaluate the top pair first, then raise 2 to 4.
Example 2
Problem: Evaluate 2↑↑4.
Substitution: 2^(2^(2^2)) = 2^16
Result: 65,536
Adding one level to the tower changes 16 to 2^16.
Example 3
Problem: Evaluate 3↑↑3.
Substitution: 3^(3^3) = 3^27
Result: 7,625,597,484,987
A three-level tower is 3^(3^3), not (3^3)^3.
Worked example, step by step
Rules, edge cases, and related ideas
Order matters
A power tower is evaluated right to left: a^(a^a) differs from (a^a)^a.
Recursive definition
T(a,h) is obtained by raising a to T(a,h−1), with T(a,0)=1 under this tool’s stated convention.
Fast growth
Tetration grows much faster than repeated multiplication or ordinary exponentiation as height increases.
Edge cases and limitations
- Height zero returns 1 by the empty-tower convention stated here.
- Base 1 gives 1 at every height.
- Base zero at height 1 gives 0, but height 2 reaches undefined 0⁰.
- Some input pairs exceed the supported numeric range and return a clear error.
- This finite calculator does not evaluate infinite towers or continuous tetration.
How this differs from a related concept
Exponentiation repeats multiplication; tetration repeats exponentiation. A height-three tower a↑↑3 is a^(a^a), while a³ is only a×a×a.
Understand the result
Tower height is a structural count, not an ordinary exponent. Increasing height by one can make the result vastly larger; the implementation therefore bounds base and height and reports unsupported towers explicitly.
Common mistakes
- Evaluating the tower left to right instead of right to left.
- Counting exponent signs instead of base copies to determine height.
- Confusing a↑↑h with a raised to h.
- Treating an infinite power tower as if a finite-height calculator could evaluate convergence.
Frequently asked questions
What is tetration in simple terms?
It is repeated exponentiation, the operation after exponentiation in the hyperoperation sequence.
Is a power tower evaluated from the top or bottom?
From the top, or right to left. Thus 2↑↑3 = 2^(2^2) = 16.
What does tower height count?
It counts repeated base values. A height-three tower contains three copies of the base.
Why does the calculator limit height?
Tetration grows extremely quickly, and some inputs exceed ordinary finite-number representation. Out-of-range towers are rejected rather than shown as overflow.
Does this calculate an infinite power tower?
No. It evaluates finite integer towers only; convergence of infinite towers is a different problem.