Exponent Calculator
Type a base and an exponent to get the power, exact when it can be, with the repeated multiplication or the root shown.
Fractions and decimals work in both boxes: 8 to the power 2/3, 2 to the power −3, 1.05 to the power 10.
35 =
243
3^5 = 243 exactly.
- Digits
- 3
How exponents work
An exponent says how many times to multiply the base by itself: 3⁵ = 3 × 3 × 3 × 3 × 3 = 243. Each step up multiplies by the base once more, so each step down divides by it, which explains the rest of the rules: 3⁰ = 1 (divide 3¹ by 3), and 3⁻² = 1/9 (divide by 3 twice more).
A fractional exponent is a root. The bottom of the fraction says which root and the top says which power: 8^(2/3) means the cube root of 8 (which is 2), squared, so 4. 16^(1/2) is √16 = 4. Roots of numbers that aren’t perfect powers are irrational: 2^(1/2) = √2 ≈ 1.414213562, and the calculator says so.
Large powers are written out in full here, because the arithmetic is done with exact big integers: 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376, all 31 digits. Raising a decimal is exact too: 1.05¹⁰ ≈ 1.6289, which is how £100 grows in ten years at 5% compound interest.
aⁿ = a × a × … × a (n times) · a⁰ = 1 · a⁻ⁿ = 1 ÷ aⁿ · a^(p/q) = (q-th root of a)^p
- Zero to the power zero is usually defined as 1 (the empty product), and programming languages follow this: in JavaScript, Math.pow(0, 0) and 0 ** 0 are both 1. Source: ECMAScript specification, Number::exponentiate.
- For real numbers, a negative base has no real power with an even root in the exponent: (−4)^(1/2) is not a real number. Source: Wikipedia, Exponentiation.
The exponent rules, checked
Each rule with a worked example, computed exactly.
| Rule | Example | Value |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ | 128 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² | 25 |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ | 729 |
| a⁰ = 1 | 7⁰ | 1 |
| a⁻ⁿ = 1 ÷ aⁿ | 2⁻³ | 1/8 |
| a^(1/n) = ⁿ√a | 27^(1/3) | 3 |
| a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) | 4 |
| (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 2³ × 5³ | 1,000 |
| (a/b)ⁿ = aⁿ/bⁿ | (2/3)⁴ | 16/81 |
| 10ⁿ: n zeros | 10⁹ | 1,000,000,000 |
Show all 11 rowsShow fewer
| (−a)ⁿ, n odd | (−2)⁵ | −32 |
Frequently Asked Questions
What is a negative exponent?
The reciprocal of the positive power: 2⁻³ = 1/2³ = 1/8. It is not a negative number; the minus sign means divide instead of multiply.
Is −3² equal to 9 or −9?
By the usual order of operations, −3² = −(3²) = −9; the exponent applies before the minus. Write (−3)² for 9. This calculator treats the base box as one number, so a base of −3 squared is 9.
What is 2 to the power 0.5?
The square root of 2, about 1.41421356. It is irrational, so the decimal never ends or repeats.