What an exponent means
An exponent says how many times to multiply a base by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent (or power) is 5. Every rule below follows from that definition.
The rules of exponents
| Rule | In symbols | Example |
|---|---|---|
| Product rule | aᵐ · aⁿ = aᵐ⁺ⁿ | 2³ · 2⁴ = 2⁷ = 128 |
| Quotient rule | aᵐ / aⁿ = aᵐ⁻ⁿ | 5⁶ / 5⁴ = 5² = 25 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x³)² = x⁶ |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x²y)³ = 8x⁶y³ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (3/4)² = 9/16 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) = (∛8)² = 4 |
Why the rules work
Product rule. 2³ · 2⁴ is three 2s times four 2s, which is seven 2s. The exponents add.
Quotient rule. In 5⁶/5⁴, four of the six 5s on top cancel with the four below, leaving 5².
Zero exponent. By the quotient rule, a³/a³ = a⁰. But anything nonzero divided by itself is 1, so a⁰ = 1.
Negative exponents. a²/a⁵ = a⁻³ by the quotient rule, and it also equals 1/a³ after cancelling. So a negative exponent means "one over". It doesn't make the number negative: 2⁻³ = 1/8 = 0.125.
Fractional exponents. Since (a^(1/2))² = a¹ = a, a^(1/2) must be √a. In general the denominator is a root and the numerator is a power: 16^(3/4) = (⁴√16)³ = 2³ = 8, and 27^(−1/3) = 1/∛27 = 1/3.
Worked examples
- Simplify (3x²)³ · x⁴. Power of a product: 3³x⁶ = 27x⁶. Product rule: 27x⁶ · x⁴ = 27x¹⁰.
- Simplify x⁵y⁻² / (x²y³). Subtract exponents base by base: x⁵⁻² y⁻²⁻³ = x³y⁻⁵ = x³/y⁵.
- Evaluate 4^(3/2). Square root first, then cube: (√4)³ = 2³ = 8.
- Evaluate (2/3)⁻². Flip and make the exponent positive: (3/2)² = 9/4.
- Write 1/√x as a power: x^(−1/2). This form is what you need to use the power rule for derivatives; see the derivative rules table.
Common mistakes
- Adding powers of a sum. (a + b)² is a² + 2ab + b², not a² + b². The rules work for products and quotients, not sums.
- Multiplying exponents in a product. x² · x³ = x⁵, not x⁶. You multiply exponents only for a power of a power.
- Different bases. 2³ · 3² can't be combined with the product rule; it's 8 × 9 = 72.
- Negative bases. −3² means −(3²) = −9, but (−3)² = 9. Brackets matter.
- Zero to the zero. 0⁰ is left undefined in many algebra courses, though some areas of math define it as 1. Calculators differ.
Scientific notation uses the same rules
Numbers in scientific notation are powers of 10, so the product and quotient rules do the heavy lifting. Multiply the front numbers and add the exponents: (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹. Divide the front numbers and subtract the exponents: (8 × 10⁶)/(2 × 10²) = 4 × 10⁴. If the front number ends up 10 or more, shift one power across: 12 × 10⁹ = 1.2 × 10¹⁰. A power of a power works too: (2 × 10³)² = 4 × 10⁶.
Exponents and logarithms
A logarithm undoes an exponent: if 2ˣ = 32, then x = log₂ 32 = 5. The product rule for exponents becomes the product rule for logs, log(ab) = log a + log b. For logs in bases your calculator doesn't have, use the change-of-base formula.
Work out any power exactly, including negative and fractional exponents, with the exponent calculator, or solve for a missing exponent with the log calculator. For very large or small numbers, see the scientific notation calculator.
Sources
- OpenStax College Algebra 2e, 1.2: Exponents and scientific notation
- OpenStax Intermediate Algebra 2e, 5.2: Properties of exponents and scientific notation
Every worked example on this page was checked independently with SymPy.