Build a power tower
A power tower repeats the same base in a vertically nested exponent. The exponentiation is evaluated from the top down.
What is a power tower?
A power tower is an expression in which exponentiation is nested several times. For example:
This expression contains three occurrences of the number 2. Exponentiation is evaluated from the top down:
The order matters. A power tower is right-associative, meaning a^(b^c) rather than (a^b)^c.
Power towers and tetration
Tetration is the operation that constructs repeated power towers. It is commonly written using double up-arrow notation:
For a positive integer height n, this represents a tower of n copies of a, evaluated from the top down.
| Notation | Expanded expression | Value |
|---|---|---|
| 2 ↑↑ 1 | 2 | 2 |
| 2 ↑↑ 2 | 2² | 4 |
| 2 ↑↑ 3 | 2^(2²) | 16 |
| 2 ↑↑ 4 | 2^(2^(2²)) | 65,536 |
The calculator uses positive integer tower heights. Its custom decimal base option is an extension for visualization, not a definition of every possible generalized tetration function.
Why do power towers grow so quickly?
Ordinary exponentiation already grows rapidly. A power tower repeats exponentiation inside the exponent, causing growth far faster than ordinary powers.
For example, the height-four tower with base 2 is:
The next tower is:
This value has more than 19,000 decimal digits. Adding just one level can make a result too large for ordinary calculator representations.
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Frequently asked questions
What does a power tower represent?
A power tower represents repeated, nested exponentiation. Each number is raised to the power of the expression above it.
How is a power tower evaluated?
Exponentiation is evaluated from the top down. For example, 2^(3^2) is evaluated as 2^9, giving 512.
Is a power tower the same as tetration?
A power tower of repeated identical bases is a finite tetration expression. Tetration is the operation that generalizes this construction to a variable height.
Why does the calculator show a symbolic result?
Some power towers exceed the range of ordinary floating-point numbers. In those cases, the calculator avoids displaying an invalid or misleading numerical result.
Can I use decimal bases?
Yes. The custom base option accepts positive decimal bases between 1 and 10. Decimal-base towers are evaluated using ordinary real exponentiation within the supported numerical range.