xⁿ Exponent Calculator
Raise any number to any power: exact results for whole-number exponents (2^100 in full), negative exponents as fractions, fractional exponents as roots.
Exponent rules
| 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 | 7⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | b^(1/n) = ⁿ√b | 8^(1/3) = 2 |
What Exponent Calculator Does
This exponent calculator raises any number to any power. For whole-number exponents it uses exact integer arithmetic, so results are digit-perfect however large they get — 2¹⁰⁰ appears in full as a 31-digit number, and 1.1⁵⁰ shows every decimal place. Negative exponents return the exact reciprocal as a fraction and its decimal value.
Fractional exponents are roots: type 1/3 for a cube root or 2/3 for the cube root of the square. Negative bases are handled correctly — an odd root of a negative number is real, (−8)^(1/3) = −2, while an even root is flagged as not a real number. A second mode solves bⁿ = y for the exponent using logarithms.
How to Use Exponent Calculator
- Enter the base, for example 2 or 1.5
- Enter the exponent — a whole number, a negative number, a decimal or a fraction like 1/3
- Read the exact result, or the decimal for roots
- Check the digit count and scientific notation for huge results
- Switch to Solve for the exponent to find n in bⁿ = y
Formula Used by Exponent Calculator
Powers and roots
bⁿ = b × b × … × b (n times) · b⁻ⁿ = 1 ÷ bⁿ · b^(p/q) = (q-th root of b)ᵖ
Worked example
27^(2/3)
- Cube root of 27 = 3
- 3² = 9
Result: 27^(2/3) = 9.
Solving for the exponent
bⁿ = y ⇒ n = log(y) ÷ log(b)
Worked example
2ⁿ = 1,024
- ln(1,024) = 6.931
- ln(2) = 0.6931
- 6.931 ÷ 0.6931
Result: n = 10.
Powers of 2 and 10
| n | 2ⁿ | 10ⁿ |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 2 | 10 |
| 8 | 256 | 100,000,000 |
| 10 | 1,024 | 10,000,000,000 |
| 16 | 65,536 | 10¹⁶ |
| 20 | 1,048,576 | 10²⁰ |
| 32 | 4,294,967,296 | 10³² |
| −1 | 0.5 | 0.1 |
| −3 | 0.125 | 0.001 |
How to Read Your Result
Order of operations
Exponents apply before a leading minus sign: −3² means −(3²) = −9, while (−3)² = 9. Enter −3 as the base here to mean (−3)ⁿ. Stacked exponents are evaluated from the top down: 2^3^2 = 2^(3²) = 2⁹ = 512, not (2³)² = 64.
Growth that surprises
Repeated multiplication grows faster than intuition expects. Doubling 30 times gives over a billion (2³⁰ = 1,073,741,824), and 1.07³⁰ ≈ 7.6 — why money growing 7% a year roughly doubles every ten years.
Limitations & Accuracy Notes
- Decimal and fractional exponents use floating point and are shown to about 15 significant digits.
- Exact results are shown up to about 20,000 digits; beyond that the size is given in scientific notation.
- Complex results (even roots of negative numbers) are not calculated.