Power Tower (Tetration) Calculator

Quick answer A power tower evaluates from the top down: 2↑↑3 means 2^(2^2) = 16, not (2^2)^2. That ordering is why towers explode so fast; one more level turns 16 into 65,536.

This power tower calculator works out the value of a stacked exponent, also called tetration, where a base is raised to itself repeatedly to a chosen height. You enter two numbers: the base a and the height n, which can be any whole number from 0 to 10. The calculator then evaluates the tower from the top down, since stacked exponents are right associative, meaning a raised to a raised to a is a raised to the power of a to the a, not the other way around. It returns the resulting value, along with the base and height you entered, so you can double check what was calculated. Because power towers grow at an extraordinary rate, even modest inputs produce enormous numbers; a base of 2 with a height of 4 already reaches 65,536, and larger combinations quickly exceed what any computer can store, in which case the result shows as too large to display rather than an incorrect figure. This makes the tool useful for students learning about repeated exponentiation, programmers checking algorithm output, or anyone curious how quickly these towers escalate compared with ordinary powers. Simply type in your base and height and the answer updates instantly as you type. A worked example, the formal definition and answers to common questions about tetration and its right-to-left evaluation order are set out below the calculator.

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Base a

Height n

Value = 16

Base2

Height3

The formula

A power tower of height n is a^(a^(a^(...))) with n copies of a, evaluated from the top down (right associative). Height 0 is defined as 1, height 1 is a, height 2 is a^a, and so on.

Worked example

For base 2, height 3: 2^(2^2) = 2^4 = 16. Height 4 gives 2^16 = 65536. For base 3, height 2: 3^3 = 27. Enter 2 and 3 to confirm.

Frequently asked questions

What is tetration?

Repeated exponentiation: a power tower of a, the next operation after multiplication and exponentiation.

Why evaluate from the top?

Stacked exponents are right associative, so a^a^a means a^(a^a), not (a^a)^a.

Why does it say too large?

Power towers explode in size. Even a modest height exceeds what a computer can store, so it is flagged rather than shown.

Who this calculator is for

This calculator is for students, programmers and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid input.
The standard algorithm is applied.
Results are rounded for display.

Formula and sources

Related calculators

Scientific Calculator: general scientific maths.
Percentage Calculator: percentages and changes.
Prime Factorisation Calculator: factor a number into primes.

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