Exponent Calculator
Type a base and an exponent to get aⁿ as an exact value, a decimal and in scientific notation. Negative exponents, fractional exponents such as 2/3 and huge integer powers are all handled, with the working explained.
Fractions are allowed in both boxes: 8 to the power 2/3, (3/4)², 2 to the power −1/2. Decimal exponents like 0.5 or 1.25 are converted to fractions.
Step-by-step explanation
Try an example
How to use the Exponent Calculator
- Enter the base (a). It can be a whole number, a decimal, a negative number or a fraction such as 3/4.
- Enter the exponent (n): for example 10, −3, 0.5 or 2/3.
- Pick how many significant digits you want in the decimal answer.
- Press Calculate, or just keep typing, to see the result, the exact value and scientific notation.
- Read the steps to see how negative and fractional exponents are turned into reciprocals and roots.
Laws of exponents
| Law | Rule | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| Power of a fraction | (a/b)ⁿ = aⁿ/bⁿ | (3/4)² = 9/16 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻⁴ = 1/16 = 0.0625 |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ | 27^(2/3) = 3² = 9 |
Worked examples
- 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024.
- 5⁻³ = 1/5³ = 1/125 = 0.008.
- 16^(3/4) = (⁴√16)³ = 2³ = 8.
- 2^1.5: 1.5 = 3/2, so 2^(3/2) = √(2³) = √8 = 2√2 ≈ 2.828427.
- (−8)^(2/3) = (∛−8)² = (−2)² = 4.
- 1.05³⁰ ≈ 4.3219, the growth factor of 5% compound interest over 30 years.
Powers of 2 and 10
| n | 2ⁿ | 10ⁿ | Name of 10ⁿ |
|---|---|---|---|
| 3 | 8 | 1,000 | thousand |
| 6 | 64 | 1,000,000 | million |
| 8 | 256 | 100,000,000 | hundred million |
| 10 | 1,024 | 10,000,000,000 | ten billion |
| 16 | 65,536 | 10¹⁶ | ten quadrillion |
| 20 | 1,048,576 | 10²⁰ | hundred quintillion |
| 32 | 4,294,967,296 | 10³² | hundred nonillion |
| −1 | 0.5 | 0.1 | tenth |
| −3 | 0.125 | 0.001 | thousandth |
Frequently asked questions
What does an exponent mean?
An exponent tells you how many times to multiply the base by itself. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent (or power) is 5.
How do you calculate a negative exponent?
A negative exponent means take the reciprocal: a⁻ⁿ = 1/aⁿ. For example 5⁻³ = 1/5³ = 1/125 = 0.008.
How do fractional exponents work?
The denominator is a root and the numerator is a power: a^(m/n) = (ⁿ√a)^m. For example 8^(2/3) = (∛8)² = 2² = 4, and 9^(1/2) = √9 = 3.
What is any number to the power of 0?
Any non-zero number raised to the power 0 equals 1. 0⁰ is usually defined as 1 in algebra and combinatorics, which is the convention this calculator uses, although some textbooks leave it undefined.
Can a negative number have a fractional exponent?
Only when the root is odd. (−27)^(1/3) = −3 because (−3)³ = −27, but (−4)^(1/2) has no real value because no real number squared gives −4.
How big can the numbers be?
Whole-number powers are computed exactly with big-integer arithmetic for results up to roughly 100,000 digits, for example every one of the 302 digits of 2^1000. Larger results are given in scientific notation using logarithms.
What is scientific notation?
It writes a number as m × 10^k with 1 ≤ m < 10. 2^64 = 18,446,744,073,709,551,616 ≈ 1.8446744073709552 × 10^19, written 1.8446744073709552e19 in E-notation.