What does a negative exponent mean?

A negative exponent represents the reciprocal (one-over) of the positive power. a^(−n) = 1 / a^n. For example, 2^(−3) = 1 / 2^3 = 1/8. This keeps the law a^b × a^c = a^(b+c) consistent for all integer exponents — in particular, a^1 × a^(−1) = a^0 = 1 forces a^(−1) = 1/a.

What does a fractional exponent mean?

A fractional exponent represents a root. a^(1/n) is the nth root of a, and a^(p/q) is the qth root of a^p (or equivalently, the qth root of a, raised to p). For example, 8^(2/3) means the cube root of 8 squared: ∛8 = 2, then 2² = 4.

Why does my large exponent show 'Infinity'?

JavaScript numbers use IEEE 754 double precision, which tops out at approximately 1.8 × 10^308. Any larger result is represented as 'Infinity'. For truly huge calculations (e.g., cryptography with 500-digit numbers), specialized big-integer libraries are needed — this calculator uses native floating-point.

Is a^b the same as b^a?

Almost always no. 2^3 = 8, but 3^2 = 9. The order of base and exponent matters, and they are rarely interchangeable. A famous exception: 2^4 = 16 = 4^2, and this is the only positive-integer pair (other than equals) with that property.

What is exponential growth, and why does it matter?

Exponential growth is when a quantity is multiplied by a constant factor per time period — compound interest, population growth, viral spread, Moore's Law. Small growth rates compound into enormous long-term changes: 1% daily growth doubles in about 70 days; 3% daily growth doubles in about 23 days. Understanding this drives intuition in finance, epidemiology, and technology forecasting.

Are exponents related to logarithms?

Yes — exponents and logarithms are inverse operations. If a^x = y, then log_a(y) = x. That is why logarithmic scales appear so naturally wherever exponential growth occurs (earthquake magnitudes, sound decibels, pH). See also the Logarithm Calculator.

Can I raise a negative number to a fractional exponent?

Only sometimes. (−8)^(1/3) = −2 is fine (cube root of a negative), but (−4)^(1/2) has no real value (square root of a negative is imaginary). Odd roots of negative numbers are real; even roots of negative numbers require complex numbers and are returned as NaN by this calculator.

Exponent / Power Calculator

Math

Calculate any base raised to any exponent (positive, negative, or decimal).

Math

Exponent / Power Calculator

Generated on April 24, 2026

Base

Exponent

?What is the Exponent / Power Calculator?

An exponent calculator computes a raised to the power b for any base a and any exponent b — integer, negative, or decimal. Exponents are fundamental to mathematics, science, engineering, and finance. They describe compound growth (money, populations, viral spread), scientific notation (10⁹ for a billion), decay (radioactive half-life), and computer science (binary sizes). Understanding exponents is essential anywhere a quantity is repeatedly multiplied by a constant factor, and this tool handles the computation regardless of whether the exponent is a whole number, a fraction (root), or a negative (reciprocal).

The Formula

a^b. If b is a positive integer, it's repeated multiplication. a^0 = 1. a^(−b) = 1 / a^b. a^(1/n) is the n-th root of a. a^(p/q) = (a^p)^(1/q).

Exponentiation generalizes repeated multiplication. Integer exponents multiply the base by itself that many times; fractional exponents represent roots (a^(1/2) is the square root); negative exponents flip to the reciprocal. All three cases follow from the laws of exponents, especially a^b × a^c = a^(b+c) and (a^b)^c = a^(b×c). These rules keep the algebra consistent across all real exponents, even irrational ones like π.

Practical Examples

2^10 = 1024 — the classic computing number (1 KiB = 1024 bytes = 2^10).

10^3 = 1000 — the meaning of 'kilo' in the metric system.

2^(−3) = 1/8 = 0.125 — negative exponent flips to a reciprocal.

8^(1/3) = 2 — the cube root of 8, a fractional exponent in action.

(1 + 0.05)^20 ≈ 2.6533 — the factor by which money grows in 20 years at 5% annual compound interest.

e^x is the exponential function that describes continuous growth — central to calculus and probability theory.

Frequently Asked Questions

Any non-zero number raised to the 0 power equals 1. This convention arises from the exponent law a^(m − n) = a^m / a^n — setting m = n gives a^0 = a^m / a^m = 1. The case 0^0 is typically defined as 1 in combinatorics and algebra, but mathematicians sometimes leave it undefined in analysis contexts.

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Exponent / Power Calculator

Exponent / Power Calculator

?What is the Exponent / Power Calculator?

The Formula

Practical Examples

Frequently Asked Questions

What is a^0?

What does a negative exponent mean?

What does a fractional exponent mean?

Why does my large exponent show 'Infinity'?

Is a^b the same as b^a?

What is exponential growth, and why does it matter?

Are exponents related to logarithms?

Can I raise a negative number to a fractional exponent?

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