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An exponent is shorthand for multiplying a number by itself repeatedly: 2⁵ means "multiply 2 by itself five times." Values grow strikingly fast as the exponent rises — from science and finance to computing and engineering, exponents are the basic way to express very large or very small numbers. Zero and negative exponents look counterintuitive at first but follow consistent rules.
Enter the base and the exponent, and the result is calculated instantly.
How is it calculated?
Definition
aⁿ = a × a × … × a (n times), where a is the base and n is the exponent. So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
Special exponents
- Zero exponent: a⁰ = 1 (for a ≠ 0). Any number to the power of zero is 1 — not by decree but as a necessary consequence of the exponent rules (aⁿ ÷ aⁿ = a⁰ = 1).
- Negative exponent: a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ = 1 ÷ 8 = 0.125. A negative exponent means "reciprocal," not a negative result.
- Exponent of one: a¹ = a.
Exponent rules (in brief)
- Same base, multiply: aᵐ × aⁿ = aᵐ⁺ⁿ
- Same base, divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Power of a power: (aᵐ)ⁿ = aᵐˣⁿ
Where it is used
- Scientific notation: very large/small numbers (3 × 10⁸, 1.6 × 10⁻¹⁹).
- Compound interest and growth: (1 + rate)ⁿ is an exponential expression.
- Computing: powers of 2 (2¹⁰ = 1,024 = 1 KB).
- Area and volume: squares (²) and cubes (³) are exponents directly.
Roots are the inverse of exponents; for an nth root use a root calculator.
Worked example
Take 2¹⁰: multiply 2 by itself ten times to get 1,024 — which is exactly why 1 kilobyte is 1,024 bytes in computing. For comparison, 5³ = 5 × 5 × 5 = 125 (the volume of a cube with side 5), and for a negative exponent 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125. The result of a negative exponent is not negative but a fraction — it means "take the reciprocal at that power." To see the power of exponents: 2¹⁰ is 1,024, but 2²⁰ exceeds one million.
FAQ
How do I calculate an exponent?+
Multiply the base by itself as many times as the exponent: 2⁵ = 2×2×2×2×2 = 32. Just enter the base and exponent; the tool supports positive, negative and zero exponents.
Why is any number to the power of zero equal to 1?+
It follows from the exponent rules: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and a number divided by itself is 1, so a⁰ = 1 (for a ≠ 0).
What does a negative exponent mean?+
a⁻ⁿ = 1 ÷ aⁿ — the reciprocal at that power. So 2⁻³ = 1 ÷ 8 = 0.125. The result is a fraction, not a negative number.
How do I raise a negative base to a power?+
An even exponent gives a positive result, an odd one negative: (−2)² = 4 but (−2)³ = −8. Parentheses matter: −2² and (−2)² give different results.
How do exponents relate to scientific notation?+
Scientific notation uses powers of 10: 300,000,000 = 3 × 10⁸. Writing very large and very small numbers as exponents makes them easier to read and compute.
What does a fractional exponent (like ½) mean?+
A fractional exponent is a root: a^(1/2) = √a, a^(1/3) = ∛a. To take roots use a root calculator; exponents and roots are inverse operations.