power to power calculator

Compute expressions like (a^b)^c or a^{(b^c)} instantly. Enter your values below and click calculate.

Base (a) The main number being raised.

First exponent (b) Exponent applied first.

Second exponent (c) Exponent applied second.

Calculation mode Order matters: exponents are not generally associative.

Decimal places (0-15) Controls formatted output precision.

What is a power to power expression?

A power to power expression is an exponent expression where exponents appear in more than one level. The two most common forms are:

(a^b)^c — a power raised to another power
a^{(b^c)} — a base raised to a powered exponent (a power tower segment)

These two forms look similar, but they are not generally equal. This is why using a dedicated calculator helps prevent mistakes.

Core formulas to remember

1) Power of a power rule

For valid values in real-number arithmetic:

(a^b)^c = a^b·c

This rule lets you multiply the exponents when a power is raised to another power.

2) Exponent stacking (right-associative)

For a tower-like form:

a^{(b^c)} means compute b^c first, then raise a to that result.

It is not the same as (a^b)^c in general.

How to use this calculator

Enter your base a.
Enter the first exponent b.
Enter the second exponent c.
Choose mode: (a^b)^c or a^(b^c).
Set desired decimal precision.
Click Calculate to see the result and step details.

Example comparisons

Example A: (2³)²

First compute 2³ = 8, then 8² = 64. Using the rule, this also equals 2⁶ = 64.

Example B: 2^(3²)

First compute 3² = 9, then 2⁹ = 512. Notice this is very different from 64 above.

Common mistakes

Assuming exponent operations are associative. They are not in general.
Ignoring parentheses. Parentheses define the order and can completely change the result.
Using negative bases with fractional exponents. Some combinations are undefined in real numbers and may return undefined/NaN.
Forgetting scale growth. Exponent expressions grow extremely fast and can overflow to Infinity.

Practical uses

Power-to-power calculations appear in many fields:

Finance: repeated compounding and nested growth models.
Physics: scaling laws and dimensional relationships.
Computer science: algorithmic complexity and cryptographic math.
Data science: transformed features and nonlinear model behavior.

FAQ

Does (a^b)^c always equal a^b·c?

In standard real-number contexts, yes for valid domains. But if intermediate steps are not defined (for example, negative base with non-integer fractional exponents), the expression can fail in real arithmetic.

Why do I sometimes see Infinity?

Because the result exceeds JavaScript’s numeric range. The calculator still reports the overflow so you know the expression grows beyond finite floating-point limits.

Can I use decimals and negative values?

Yes. The calculator supports them. If your combination is outside real-number rules, it will show an explanatory message.