Exponent Calculator
Calculate any base raised to any power. Handles large exponents, fractional powers (roots), and negative exponents.
Did this tool work for you?
How to use this calculator
Raise the base to the given exponent. Fractional exponents represent roots: x^(1/n) = ⁿ√x.
- 1
Enter the base number.
- 2
Enter the exponent (power to raise the base to).
- 3
Use fractional exponents for roots: 0.5 for square root, 0.333 for cube root.
- 4
Use negative exponents for reciprocals: 2^(-3) = 1/8.
Frequently asked questions
What does a negative exponent mean?
A negative exponent means "take the reciprocal." 2^(-3) = 1/(2^3) = 1/8 = 0.125. 10^(-2) = 1/100 = 0.01. Negative exponents appear in scientific notation: 3.5 × 10^(-6) = 0.0000035.
What does a fractional exponent mean?
x^(1/n) = the nth root of x. x^(1/2) = √x; x^(1/3) = ∛x. More generally, x^(m/n) = (ⁿ√x)^m. So 8^(2/3) = (∛8)² = 2² = 4.
What is x^0?
Any non-zero base raised to the power 0 equals 1: x^0 = 1. This is consistent with the rule x^n / x^n = x^(n-n) = x^0 = 1. The value 0^0 is mathematically indeterminate, though it is often defined as 1 in combinatorics, and that is the value this tool shows.
What is 0 to a negative power?
It is undefined. A negative exponent means a reciprocal, so 0^(-2) would be 1 ÷ 0², which is a division by zero. The tool shows "Undefined (division by zero)" for 0 raised to any negative exponent, and the same message on the reciprocal line when the base is 0 and the exponent is positive. An emptied box counts as 0, so clearing the base gives 0^exponent.
How do I calculate very large powers?
For very large exponents, results quickly become astronomical. 2^64 ≈ 1.84 × 10^19 (more than all grains of sand on Earth). At some point, the result exceeds JavaScript's floating-point precision and is shown in scientific notation.
Exponents and powers explained
Rules of exponents
x^a × x^b = x^(a+b); x^a / x^b = x^(a−b); (x^a)^b = x^(ab); (xy)^a = x^a × y^a; x^0 = 1; x^(-n) = 1/x^n; x^(1/n) = ⁿ√x. These rules are fundamental to algebra, calculus, and scientific notation.
Exponential growth in everyday life
Moore's Law: transistor counts doubled roughly every 2 years. COVID-19 early spread: exponential until interventions. Compound interest: exponential growth of money. Exponential functions grow faster than any polynomial — the fundamental insight behind why doubling times matter in epidemics and investment.
Scientific notation uses exponents of 10
Scientific notation expresses numbers as a × 10^n where 1 ≤ a < 10. Distance to Alpha Centauri: 4.07 × 10^13 km. Mass of a proton: 1.67 × 10^(-27) kg. This notation makes very large and small numbers readable and arithmetic on them practical.
Powers of 2 you will meet often
Powers of 2 appear everywhere in computing, from memory sizes to the number of values a given number of bits can hold.
| Expression | Value |
|---|---|
| 2³ | 8 |
| 2¹⁰ | 1,024 |
| 2¹⁶ | 65,536 |
| 2³² | 4,294,967,296 |
| 2⁶⁴ | about 1.8447 × 10¹⁹ (shown as 1.844674e+19) |
Negative, zero and fractional exponents
A negative exponent means a reciprocal, a zero exponent gives 1, and a fractional exponent is a root: x^0.5 is the square root and x^0.25 is the fourth root. All of these can be entered directly.
| Expression | Meaning | Result |
|---|---|---|
| 2^−3 | 1 ÷ 2³ | 0.125 |
| 5^0 | any non-zero base to the power 0 | 1 |
| 9^0.5 | square root of 9 | 3 |
| 81^0.25 | fourth root of 81 | 3 |
| 4^1.5 | 4 × √4 | 8 |
| 10^−6 | 1 ÷ 1,000,000 | 0.000001 |
Worked example: exponential growth
A bacterium that doubles every 20 minutes becomes 2¹² = 4,096 bacteria after 4 hours (12 doublings). The same maths applies to money: at 5% growth per year, 1.05¹⁴ = 1.9799, so an investment roughly doubles in 14 years.
Limits and common mistakes
A negative base with a fractional exponent has no real result, and the calculator shows NaN: work out the root of the positive number and apply the sign yourself (for the cube root of −8, calculate 8 and add the minus sign). Fractions such as 1/3 must be typed as decimals, so 0.333 is not exactly one third: 27^0.333 = 2.9967, and 27^0.3333333333 = 2.9999999997, both close to 3 but not exact. Writing 0^0 is undefined in general, and 0 raised to a negative power is a division by zero, so the calculator labels it undefined instead of giving a number.
Learn more from an authoritative source:
WikipediaLinear Equation Solver
Solve a system of two linear equations with two unknowns using substitution or Cramer's rule.
Modulo Calculator
Calculate the remainder of division (modulo operation). Supports positive and negative numbers and shows the full division breakdown.
Rectangle Calculator
Calculate area, perimeter, diagonal, and aspect ratio of a rectangle or square.
Mean, Median & Mode Calculator
Calculate mean, median, mode, range, and variance for a set of numbers. Enter up to 20 values.
Number Sequence Calculator
Identify and continue arithmetic and geometric sequences. Find the nth term and sum of the first n terms.
Roman Numerals Converter
Convert between Roman numerals and Arabic (standard) numbers. Handles values from 1 to 3,999.
Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →