An exponent tells you how many times to use a number as a factor: 2¹⁰ means ten 2s multiplied together, which is 1,024. The idea stretches further than repeated multiplication, though. Negative exponents produce reciprocals, fractional exponents produce roots, and decimal exponents fill in everything in between. This calculator handles all of those cases and keeps the answer exact whenever it can.
How to use the exponent calculator
- Enter the base b. Negative numbers, decimals and fractions like 3/4 are fine.
- Enter the exponent n, for example 10, −3, 0.5 or 2/3.
- The tape shows the exact value (or a simplified radical), a decimal and scientific notation where useful. The steps name the exponent rule that applies.
Exponent rules
| Rule | Formula | Example |
|---|---|---|
| Product | bᵐ · bⁿ = bᵐ⁺ⁿ | 2³ · 2⁴ = 2⁷ = 128 |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (3²)³ = 3⁶ = 729 |
| Zero exponent | b⁰ = 1 (b ≠ 0) | 7⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 10⁻³ = 0.001 |
| Fractional exponent | b^(p/q) = (q-th root of b)ᵖ | 27^(2/3) = 9 |
The zero and negative exponent rules are not arbitrary: they are the only values that keep the quotient rule true. Since 5⁴ ÷ 5⁴ must equal 1 and the rule says it equals 5⁰, 5⁰ has to be 1.
Worked examples
Whole exponent: 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024.
Negative exponent: 5⁻³ = 1 ÷ 5³ = 1 ÷ 125 = 0.008.
Fractional exponent: 8^(2/3) = (∛8)² = 2² = 4.
Negative base, odd root: (−27)^(−2/3) = (∛−27)⁻² = (−3)⁻² = 1/9.
Decimal exponent: 1.05¹² = 1.795856326…, the growth factor for twelve periods of 5% growth.
For an exponent such as 0.123456 that is not a simple fraction, the calculator uses logarithms: b^n = 10^(n·log₁₀ b). That is how 2^0.123456 ≈ 1.0893412746 is found.
Where exponents show up
- Compound growth. An amount growing 5% a year for 12 years is multiplied by 1.05¹², about 1.796, so it grows roughly 80%.
- Scientific notation. Powers of ten describe very large and very small quantities: Avogadro’s number is about 6.022 × 10²³ and a nanometer is 10⁻⁹ m.
- Computing. Memory sizes are powers of two: 2¹⁰ = 1,024 bytes in a kibibyte and 2³² ≈ 4.29 billion possible IPv4 addresses.
- Area and volume. Squaring and cubing a length (x² and x³) give area and volume, which is why doubling a cube’s side multiplies its volume by 8.
Common mistakes
- Multiplying base and exponent. 3⁴ is 81, not 12.
- Distributing over addition. (a + b)² is a² + 2ab + b², not a² + b².
- Sign confusion. A negative exponent gives a small positive number, not a negative one: 2⁻³ = 0.125.
- Even roots of negatives. (−16)^(1/2) has no real value; (−16)^(1/4) does not either.
When the result is a whole number with hundreds or thousands of digits, the large exponents calculator prints every digit. To go the other way and find the exponent, use the solve for exponent calculator or the logarithm calculator.
Frequently asked questions
What does a negative exponent mean?
It means a reciprocal: b to the −n equals 1 divided by b to the n. For example 5⁻³ = 1/5³ = 1/125 = 0.008. A negative exponent never makes the answer negative by itself.
What does a fractional exponent mean?
The denominator is a root and the numerator is a power: b^(p/q) = (q-th root of b)^p. So 8^(2/3) = (∛8)² = 2² = 4, and 2^0.5 = √2 ≈ 1.4142.
Why is (−8)^(1/3) a real number but (−4)^(1/2) is not?
Odd roots of negative numbers exist (−2 × −2 × −2 = −8), but no real number squares to −4. So a negative base has a real fractional power only when the reduced denominator of the exponent is odd. Otherwise the calculator reports the principal complex value.
What is 0 to the power 0?
Algebra, combinatorics and most software define 0⁰ = 1 so formulas such as the binomial theorem work without exceptions. In calculus, 0⁰ is called indeterminate when it appears as a limit, because different limits approach different values.
Is −2² the same as (−2)²?
No. Exponents are applied before the minus sign, so −2² = −(2²) = −4 while (−2)² = 4. In this calculator a negative base is always treated as if it were in parentheses.