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
| Field | What to enter |
|---|---|
| Find | Choose one: Result a = (bᵐ)ⁿ, Base b, Exponent m, Exponent n |
| Base b | A number |
| Inner exponent m | A number |
| Outer exponent n | A number |
| Result a | A 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 b | Result (bᵐ)ⁿ |
|---|---|
| 1 | 1 |
| 1.5 | 11.3906 |
| 2 | 64 |
| 2.5 | 244.1406 |
| 3 | 729 |
| 4 | 4,096 |
Formula
(bᵐ)ⁿ = b^(m·n)
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.
Frequently asked questions
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.