Quick answer: An exponent says how many times to multiply the base by itself: 2^10 = 2 × 2 × … × 2 (ten times) = 1,024.
How it's calculated
For a whole exponent n, aⁿ is a multiplied by itself n times, and a⁰ = 1 for any a other than 0.
A negative exponent means the reciprocal: a^(−n) = 1 ÷ aⁿ, so 5^(−2) = 1 ÷ 25 = 0.04. A fractional exponent is a root: a^(p/q) is the q-th root of a raised to p, so 27^(1/3) = 3 and 8^(2/3) = 4.
A negative base works with whole exponents and with fractions whose denominator is odd: (−8)^(1/3) = −2. With an even denominator, such as (−4)^(1/2), there is no real answer. For roots on their own use the root calculator, and for exact powers of large whole numbers the big number calculator.
Worked examples
- Whole exponent: 2^10 = 1,024
- Negative exponent: 5^−2 = 1 ÷ 25 = 0.04
- Fractional exponent: 8^(2/3) = (∛8)² = 2² = 4
Exponent rules
| Rule | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| a⁰ = 1 | 7⁰ = 1 |
| a⁻ⁿ = 1 ÷ aⁿ | 10⁻³ = 0.001 |
Frequently asked questions
What is any number to the power 0?
1, for any base other than 0. 0⁰ is usually taken as 1 in algebra and counting, and this calculator follows that convention.
How do I enter a cube root as an exponent?
Type 1/3 as the exponent: 125^(1/3) = 5.
Why is (−2)^0.5 not allowed?
0.5 is 1/2, a square root, and the square root of a negative number is not a real number.
What is 2 to the power 10?
1,024, which is why a kilobyte was traditionally 1,024 bytes.
What is 10 to the power −3?
0.001, one thousandth.
What are the rules for combining powers?
With the same base, multiplying adds the exponents and dividing subtracts them. Raising a power to another power multiplies them, so (2³)² = 2⁶ = 64.