# Power of a Power Calculator

URL: https://speedycalculator.com/calculators/math/power-of-a-power/
Last updated: 2026-10-30

> The power of a power calculator simplifies (bᵐ)ⁿ = bᵐⁿ and finds the value – or solves for the base or either exponent when the result is known.

## Inputs

- Find
- Base b
- Inner exponent m
- Outer exponent n
- Result a

## Results

- Result (bᵐ)ⁿ
- Combined exponent m·n
- Base b
- Exponent m
- Exponent n

## Formula

```
(bᵐ)ⁿ = b^(m·n)
```

## Explanation

Raising a power to another power multiplies the exponents: (2³)² = 2⁶ = 64. The rule is used to simplify algebra, to rewrite roots of powers and in scientific notation.

### How to use the Power of a Power Calculator

- **Find**
- **Base b**
- **Inner exponent m**
- **Outer exponent n**
- **Result a**

Results update as you type, and everything is calculated in your browser.

### How it works

Choose which value is unknown and enter the others.

### Inputs explained

FieldWhat to enterFindChoose one: Result a = (bᵐ)ⁿ, Base b, Exponent m, Exponent nBase bA numberInner exponent mA numberOuter exponent nA numberResult aA number

### What each result means

- **Result (bᵐ)ⁿ** = ([Base b]^[Inner exponent m])^[Outer exponent n]
- **Combined exponent m·n** = [Inner exponent m]×[Outer exponent n]
- **Base b** = [Result a]^(1÷([Inner exponent m]×[Outer exponent n]))
- **Exponent m** = ln([Result a])÷([Outer exponent n]×ln([Base b]))
- **Exponent n** = ln([Result a])÷([Inner exponent m]×ln([Base b]))

### Example values

How the main result changes when **Base b** changes and everything else stays at the example values above:

Base bResult (bᵐ)ⁿ111.511.39062642.5244.1406372944,096

## Worked example

Inputs — Find: Result a = (bᵐ)ⁿ; Base b: 2; Inner exponent m: 3; Outer exponent n: 2.

Results — **Result (bᵐ)ⁿ = 64**; Combined exponent m·n = 6.

## Key terms

- **Power rule**: (bᵐ)ⁿ = bᵐⁿ.

## FAQ

**What is (x²)³?**
x⁶.

**Is (bᵐ)ⁿ the same as b^(mⁿ)?**
No – (2³)² = 64 but 2^(3²) = 512.

**Does the order of m and n matter?**
No – (bᵐ)ⁿ = (bⁿ)ᵐ.

**How does it apply to roots?**
√(x⁶) = (x⁶)^(1/2) = x³.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)