Calculate a Power Tower
Choose a repeated base and tower height, or enter a custom sequence of bases to evaluate a more general power tower.
How to Use the Power Tower Calculator
1. Select the mode
Choose repeated base mode for ordinary power towers or custom mode for different bases at each level.
2. Enter the values
Specify the base and height, or enter a sequence of bases separated by commas.
3. Calculate
Submit the form to evaluate the power tower and inspect the calculation steps.
What Is a Power Tower?
A power tower is a mathematical expression consisting of multiple exponentiations arranged vertically or written as nested powers. Each exponent is itself another power, creating a structure that can grow extremely quickly.
Power towers are closely related to tetration, which is the fourth operation in the conventional hyperoperation sequence. Tetration generalizes exponentiation by repeatedly applying the operation of exponentiation.
For example, a power tower with base 2 and height 3 is:
Notice that exponentiation is evaluated from right to left. Therefore, a power tower is not evaluated in the same way as an ordinary left-to-right sequence of operations.
Power Tower Formula
A repeated-base power tower of height h can be represented recursively:
T(a, h) = a ^ T(a, h - 1)
Here, a represents the base and h represents the number of occurrences of that base in the tower.
Examples of Power Towers
| Base | Height | Expression | Result |
|---|---|---|---|
| 2 | 2 | 2 ^ 2 | 4 |
| 2 | 3 | 2 ^ (2 ^ 2) | 16 |
| 2 | 4 | 2 ^ (2 ^ (2 ^ 2)) | 65,536 |
| 3 | 2 | 3 ^ 3 | 27 |
| 3 | 3 | 3 ^ (3 ^ 3) | 7,625,597,484,987 |
Power Towers and Tetration
Tetration is the mathematical operation that describes repeated exponentiation. A power tower with a repeated base is a direct representation of tetration.
The notation a ↑↑ h is commonly used to represent tetration with base a and height h. The exact notation conventions can vary between mathematical references.
Why Do Power Towers Grow So Quickly?
Exponential expressions already grow much faster than ordinary polynomial expressions. Power towers take this further by placing exponentials inside other exponentials. Even relatively small bases and heights can therefore produce enormous values.
This rapid growth makes power towers useful in studying large numbers, advanced number theory, hyperoperations and mathematical growth rates.
Calculation Limitations
This calculator uses PHP floating-point arithmetic. Results are therefore approximations for many non-integer calculations and may be displayed in scientific notation. Extremely large towers can exceed the available numerical range. The calculator reports such overflow rather than presenting an incorrect finite answer.
Only positive real bases and integer tower heights are supported. For bases between zero and one, underflow can occur when results become too small for floating-point representation.
Related Mathematical Calculators
Explore other mathematical tools on Summe.org related to exponentiation, tetration, hyperoperations and mathematical sequences.
Frequently Asked Questions
1. What is a power tower calculator?
A power tower calculator evaluates nested exponentiation expressions. It allows users to specify a base and tower height and calculates the resulting value.
2. How is a power tower evaluated?
A power tower is evaluated from the top down, or equivalently from the bottom of its written expression upwards. Exponentiation is right-associative.
3. What is the difference between exponentiation and tetration?
Exponentiation raises one number to another power. Tetration repeats exponentiation to create a power tower.
4. Can I calculate power towers with different bases?
Yes. Select custom sequence mode and enter up to eight positive bases separated by commas. For example, 2, 3, 4 evaluates as 2^(3^4).
5. Why do power tower results become so large?
Each level introduces another exponentiation. This causes power towers to grow much faster than ordinary exponential expressions.
6. Can this calculator evaluate infinitely tall towers?
No. This calculator evaluates finite towers with heights from 1 to 8. Infinite power towers require separate convergence analysis.