The short answer
An exponent tells you how many times to multiply a base by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Negative exponents flip to a reciprocal (a⁻ⁿ = 1/aⁿ), and fractional exponents are roots in disguise (a^(1/2) = √a). This calculator solves both directions — powers and roots — for any base.
Key takeaways
- A negative exponent doesn't make the result negative — it makes the result a reciprocal: 2⁻³ = 1/2³ = 1/8, not -8.
- Any nonzero number raised to the power of 0 equals 1, including negative bases and fractions — but 0⁰ is undefined.
- Fractional exponents combine a power and a root in one step: a^(m/n) = ⁿ√(aᵐ).
- Even roots of negative numbers (square root, 4th root) aren't real numbers; odd roots of negative numbers (cube root, 5th root) are.
The core exponent rules
| Rule | Formula | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ = 32 |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 5⁰ = 1 |
| Fractional exponent | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = 4 |
Worked example: a fractional exponent
8^(2/3) = (∛8)² — take the root first
∛8 = 2
2² = 4
The denominator of the fractional exponent (3) is the root, and the numerator (2) is the power — you can apply them in either order. Taking the root first, as shown above, usually keeps the numbers smaller and easier to work with than raising to the power first (∛(8²) = ∛64 = 4 gives the same answer, but 64 is a bigger intermediate number than 2).
Common mistakes to avoid
- Treating a negative exponent as if it makes the result negative — 2⁻³ equals 1/8, a positive fraction, not -8.
- Confusing (-2)² with -2² — (-2)² = 4, but -2² = -4, since without parentheses the exponent applies before the negation.
- Expecting a real result from an even root of a negative number — (-16)^(1/4) has no real-number answer.
- Multiplying the base by the exponent instead of exponentiating — 2⁵ is 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10.
Related calculators
- Root Calculator — focus specifically on square roots, cube roots, and nth roots.
- Log Calculator — solve the inverse problem: what exponent produces a given result.
- Scientific Notation Calculator — see powers of 10 at work in very large or small numbers.
- Big Number Calculator — raise numbers to large integer powers without losing precision.