Exponent Calculator
Raise any number to any power, including negative and fractional exponents, with the equivalent root and reciprocal forms shown.
How to use this calculator
- 1Enter the base and the exponent.
- 2Use a negative exponent for reciprocals and a fractional one for roots.
- 3The nearby-powers table shows how quickly the value grows or shrinks.
How the calculation works
xⁿ = x × x × … × x (n times) x⁻ⁿ = 1 / xⁿ x^(1/n) = ⁿ√x- x
- The base — the number being multiplied
- n
- The exponent — how many times, or which root
A negative exponent is the reciprocal of the positive power, not a negative result.
A fractional exponent is a root: x^(1/2) is the square root, x^(1/3) the cube root.
x⁰ = 1 for any non-zero x. 0⁰ is left undefined here, as it is context-dependent in mathematics.
Worked example
2 to the power of 10
- 1.2¹⁰ means 2 multiplied by itself ten times.
- 2.2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024.
- 3.This is why a kibibyte is 1,024 bytes rather than 1,000 — computers count in powers of two.
Result: 1,024
What raising a number to a power really means
An exponent is repeated multiplication written compactly: 2⁵ means 2 multiplied by itself five times, rather than five written out longhand. The exponent tells you how many copies of the base to multiply together, which is why exponents grow so much faster than simple multiplication — each additional exponent does not add another copy, it multiplies the running total by the base all over again.
Exponential growth and decay show up everywhere
Once a quantity’s rate of change depends on how much of it already exists, the mathematics involved is exponential.
- Compound interest — a savings balance earning interest on its own accumulated interest grows exponentially, not in a straight line.
- Computing — memory, storage and processing power are conventionally measured and priced in powers of two, from a single bit to a kibibyte’s 1,024 bytes.
- Population and epidemics — both unrestricted population growth and the early spread of an outbreak follow the same exponential curve, just with a growth factor above or below 1.
- Physics and engineering — radioactive decay, capacitor discharge and sound attenuation are all exponential-decay processes, built on a base between 0 and 1.
Negative and fractional exponents, demystified
A negative exponent does not make the result negative — it means "take the reciprocal." 2⁻³ is 1 ÷ 2³, which comes out to a small positive fraction, not −8. A fractional exponent, meanwhile, is a root in disguise: raising to the power 1/2 is exactly the same operation as taking a square root, and 1/3 is a cube root, because a fractional power and a root are simply two ways of writing the same inverse operation.
What this assumes, and where it stops
Assumptions
- Real-number arithmetic only. Complex results are rejected rather than approximated.
Limitations
- A negative base with a fractional exponent produces a complex number and is rejected.
- Results beyond about 1.8 × 10³⁰⁸ exceed double-precision range and are rejected.
- Very large exponents lose precision in the final digits, as all floating-point arithmetic does.
Common questions
What does a negative exponent mean?
It means the reciprocal, not a negative number. 2⁻³ is 1 ÷ 2³ = 1/8 = 0.125. The result stays positive whenever the base is positive — the minus sign applies to the operation, not the value.
Why is anything to the power of zero equal to 1?
Because of the division rule. Since xᵃ ÷ xᵇ = x^(a−b), setting a = b gives xᵃ ÷ xᵃ = x⁰. Anything divided by itself is 1, so x⁰ must be 1 for any non-zero x.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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