Exponent Calculator
Enter a base and an exponent to get the exact power: whole-number results to the last digit, fractions for negative exponents, exact roots when they exist and decimals when they do not. The working is written out, and a table checks the exponent laws on your own numbers. Free, no sign-up.
A whole number, decimal or fraction such as 2, −1.5 or 3/4.
Whole, negative, decimal or fraction: 10, −3, 0.5 or 2/3.
Used only by the laws table below; whole numbers.
Results
1024
Result
4
Digits in the result
The working
- Write the exponent 10 in binary: 1010. That means 10 = 2 + 8.
- Square repeatedly: 2¹ = 2, 2² = 4, 2⁴ = 16, 2⁸ = 256.
- Multiply the powers that match the binary digits: 2² × 2⁸ = 1024.
The exponent laws on your numbers
With your base and the two whole-number exponents, both sides of each law come out the same.
| Law | Rule | With your numbers |
|---|---|---|
| Product rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2¹⁰ × 2³ = 1024 × 8 = 8192; 2¹³ = 8192 |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2¹⁰ ÷ 2³ = 1024 ÷ 8 = 128; 2⁷ = 128 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2¹⁰)³ = 1024³ = 1073741824; 2³⁰ = 1073741824 |
| Negative exponent | a⁻ⁿ = 1 / aⁿ | 2⁻³ = 0.125; 1 / 2³ = 1 / 8 = 0.125 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 2⁰ = 1 |
How to Use the Exponent Calculator
- 1. Enter the base. A whole number, a decimal or a fraction: 2, −1.5 or 3/4. To raise a negative number to a power, put the minus sign in the base — entering −2 computes (−2)ⁿ.
- 2. Enter the exponent. Whole numbers (10), negative ones (−3), decimals (0.5) and fractions (2/3) all work. Decimals are read as exact fractions, so 0.5 is 1/2.
- 3. Read the result. Whole-number powers appear in full, with a digit count and, for large values, scientific notation. Negative exponents give a fraction and a decimal; roots give an exact answer when one exists, otherwise a decimal to the places you choose.
- 4. Follow the working and the laws. The steps below show how the power is built, and the laws table checks the product, quotient and power rules on your base.
What an Exponent Means
For a whole number n ≥ 1, bⁿ means b multiplied by itself n times: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Every other kind of exponent is defined so that the same rules keep working. A zero exponent gives 1 (for b ≠ 0), because b⁰ = bⁿ ÷ bⁿ = 1. A negative exponent is a reciprocal: b⁻ⁿ = 1 / bⁿ, so 2⁻³ = 1/8 = 0.125. A fractional exponent is a root and a power: b^(p/q) is the q-th root of b raised to the power p, so 8^(2/3) = (∛8)² = 2² = 4.
| Law | Rule | Example |
|---|---|---|
| Product | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁷ ÷ 2³ = 2⁴ = 16 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 3)² = 2² × 3² = 36 |
| Negative exponent | b⁻ⁿ = 1 / bⁿ | 2⁻³ = 1/8 |
| Zero exponent | b⁰ = 1 | 2⁰ = 1 |
Worked Example: 2¹³ by Repeated Squaring
For larger whole-number exponents the calculator shows how a computer, or a patient person, builds the power with few multiplications. Write the exponent in binary, square the base repeatedly, and multiply the squares that match the 1 digits.
- 13 in binary is 1101, so 13 = 1 + 4 + 8.
- Squaring repeatedly: 2¹ = 2, 2² = 4, 2⁴ = 16, 2⁸ = 256.
- Multiply the matching powers: 2¹ × 2⁴ × 2⁸ = 2 × 16 × 256 = 8192.
- Three squarings and two multiplications replace twelve one-at-a-time multiplications.
Very Large Results
Whole-number powers are computed with exact big-integer arithmetic, so there is no rounding at any size: 2¹⁰⁰ is 1267650600228229401496703205376 to the last digit, which floating-point arithmetic can only approximate as 1.2676506002282294 × 10³⁰. Results with up to 20,000 digits are written out in full; beyond that, the calculator gives an order-of-magnitude estimate from logarithms (2¹⁰⁰⁰⁰⁰ ≈ 9.99002093 × 10³⁰¹⁰²) and says so.
Conventions to Know
- 0⁰ = 1. This is the convention in combinatorics, power series and most software; some analysis texts leave it undefined, and the calculator flags the case.
- Zero to a negative power is undefined, since it would mean dividing by zero.
- Negative bases with fractional exponents. The exponent is reduced to lowest terms first. If its denominator is odd, the real root is used, so (−8)^(1/3) = −2 and (−8)^(2/3) = 4. If it is even, there is no real answer: (−4)^(1/2) is the complex number 2i.
- −2² and (−2)² differ. Typed as an expression, −2² means −(2²) = −4. Here the base is a number, so entering −2 with exponent 2 gives 4.
Frequently Asked Questions
- What is a negative exponent?
- A negative exponent means a reciprocal, not a negative result. 5⁻² = 1/5² = 1/25 = 0.04. The sign of the answer depends only on the base, so (−2)⁻³ = −1/8.
- What does a fractional exponent mean?
- The denominator is the root and the numerator is the power. 27^(2/3) is the cube root of 27, which is 3, squared, giving 9. An exponent of 1/2 is a square root, so 9^0.5 = 3.
- Why is any number to the power zero equal to 1?
- Because the quotient rule requires it: bⁿ ÷ bⁿ = bⁿ⁻ⁿ = b⁰, and anything nonzero divided by itself is 1. The case 0⁰ is a separate convention, usually taken as 1.
- Is 2³² the same as (2³)²?
- No. A stacked exponent is evaluated from the top down, so 2^(3²) = 2⁹ = 512, while (2³)² = 8² = 64. To get the first, enter 9 as the exponent; for the second, enter 6.
- Does (a + b)² equal a² + b²?
- No. (a + b)² = a² + 2ab + b². Exponent laws apply to products, like (ab)², but not to sums. For a = 3 and b = 4, (3 + 4)² = 49, while 3² + 4² = 25.
- How do I find the exact value of a very large power?
- Use big-integer arithmetic rather than a floating-point calculator. This tool does that for whole-number exponents and shows every digit, up to a 20,000-digit result.