Exponent Calculator
Compute any power, including negative and fractional exponents, with the laws shown.
2 multiplied by itself 10 times.
An exponent is repeated multiplication: aⁿ means multiply a by itself n times. So 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024. The base a is the number being multiplied and the exponent n counts the factors. A negative exponent gives a reciprocal and a fractional exponent gives a root.
What an exponent is
In the expression aⁿ, the base a is the number you multiply and the exponent n tells you how many times to use it as a factor. So 2³ = 2 × 2 × 2 = 8, and 5² = 5 × 5 = 25. Exponents are the engine behind scientific notation, compound growth, and area and volume formulas, because they compress long repeated products into a single compact symbol.
the product rule — multiplying powers of the same base adds their exponents
Worked example
Evaluate 2¹⁰ by repeated multiplication:
- 1 Identify base and exponent. In 2¹⁰ the base is 2 and the exponent is 10.
- 2 Write the repeated product. 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (ten 2s).
- 3 Multiply step by step. 2 → 4 → 8 → 16 → 32 → 64 → 128 → 256 → 512 → 1024.
- 4 Read the result. 2¹⁰ = 1024. ✓
The laws of exponents
Same-base rules that turn powers into simple arithmetic on the exponents.
| Law | Rule | Example |
|---|---|---|
| Product | aᵐ · aⁿ = aᵐ⁺ⁿ | 2³ · 2² = 2⁵ = 32 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1 ÷ aⁿ | 2⁻³ = 1 ÷ 8 = 0.125 |
| Fractional exponent | a^(1 ÷ n) = ⁿ√a | 9^(1 ÷ 2) = √9 = 3 |
Negative, fractional, and the 0⁰ case
Negative exponents are reciprocals. A negative exponent flips the base into the denominator: a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125. This keeps the product rule consistent, since aⁿ · a⁻ⁿ = a⁰ = 1.
Fractional exponents are roots. The denominator of a fractional exponent is a root: a^(1 ÷ n) = ⁿ√a, and a^(m ÷ n) = ⁿ√(aᵐ). So 8^(1 ÷ 3) = ³√8 = 2 and 4^(3 ÷ 2) = (√4)³ = 8. A negative base with a fractional exponent, such as (−4)^(1 ÷ 2), has no real value.
The 0⁰ convention. 0⁰ is an indeterminate form — different limits give different answers — but in algebra and combinatorics it is almost always defined as 0⁰ = 1, which this calculator follows. Meanwhile 0 raised to a negative power is genuinely undefined, because it would require dividing by zero.