An exponent calculator with exact answers and the rules shown
This free exponent calculator raises any number to any power. Whole-number powers are computed exactly with big-integer arithmetic, so 2¹⁰⁰ shows all 31 digits and 7¹⁰⁰⁰ shows all 846 — not a rounded 1.27 × 10³⁰. Negative exponents give exact fractions (2⁻³ = 1/8), and fractional or decimal exponents such as 8^(2/3) or 4^0.5 are turned into roots and powers step by step.
Two extra modes solve the equation the other way round: find the exponent in aˣ = c (a logarithm) or the base in xᵇ = c (a root). It is built for algebra homework, computer-science powers of two, compound-growth questions and anyone curious just how big 9⁹⁹ really is.
How to use the Exponent Calculator
- Choose Power aᵇ, Find the exponent or Find the base.
- Type the base. Decimals and negative numbers are fine, for example 1.05 or −3.
- Type the exponent as a whole number, a negative number, a decimal (0.25) or a fraction (3/4).
- Read the answer in the result panel. Long exact answers appear in full in the box below, with a button to copy every digit.
- Check the digit count and scientific notation in the tiles, and follow the numbered working to see which exponent rule was used.
Exponent Calculator features
Exact big results
Integer powers up to 100,000 digits are calculated exactly; beyond that you get the digit count and scientific notation.
Negative exponents as fractions
5⁻² is shown as the exact fraction 1/25 and the decimal 0.04, not just a rounded number.
Fractional and decimal powers
0.75 is converted to 3/4 and evaluated as a root then a power, with exact answers whenever the root is exact.
Negative bases done right
Odd roots of negatives are allowed — (−8)^(1/3) = −2 — while impossible cases such as (−4)^0.5 are explained.
Solve for the exponent or base
Find x in 2ˣ = 1024 or in x³ = 27, with logarithm or root steps.
Laws of exponents
Every step this calculator shows comes from a handful of rules. They hold for any non-zero base a and b, and any exponents m and n.
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | 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 quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)² = 4/9 |
| Zero exponent | a⁰ = 1 | 17⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) = 2² = 4 |
Worked examples
- Negative exponent: 10⁻⁴ = 1/10⁴ = 1/10,000 = 0.0001. A negative exponent never makes the answer negative — it makes it a reciprocal.
- Fractional exponent: 32^(3/5) = (⁵√32)³ = 2³ = 8. Take the root first; the numbers stay small.
- Decimal exponent: 16^0.75 = 16^(3/4) = (⁴√16)³ = 2³ = 8.
- Growth: money growing 8% a year for 10 years is multiplied by 1.08¹⁰ ≈ 2.1589, so ₹50,000 becomes about ₹1,07,946. The Compound Interest Calculator handles deposits and compounding periods.
- Powers of two: 2¹⁰ = 1,024 (a kibibyte), 2³² = 4,294,967,296 (the number of IPv4 addresses) and 2⁶⁴ has 20 digits.
Common mistakes with exponents
- −3² vs (−3)². Without brackets the power applies only to 3, so −3² = −9, while (−3)² = 9. This calculator always treats the base you type as a whole, so −3 with exponent 2 gives 9.
- Multiplying the base by the exponent. 5³ is 5 × 5 × 5 = 125, not 5 × 3 = 15.
- Adding exponents across different bases. 2³ × 3² cannot be combined into one power; the product rule needs the same base.
- Distributing over a sum. (a + b)² = a² + 2ab + b², not a² + b².
Exponent Calculator FAQ
How do I calculate a number to a power?
Multiply the base by itself as many times as the exponent says: 4³ = 4 × 4 × 4 = 64. For large or awkward exponents, type the base and exponent into the exponent calculator and it works the answer out exactly.
What does a negative exponent mean?
It means “one over” the positive power: a⁻ⁿ = 1/aⁿ. So 3⁻² = 1/9 ≈ 0.111. The calculator shows both the exact fraction and the decimal.
How do fractional exponents work?
The denominator is a root and the numerator is a power: a^(m/n) = (ⁿ√a)ᵐ. For example 27^(2/3) = (∛27)² = 3² = 9. Decimal exponents are converted to fractions first, so 0.5 means a square root.
Why is any number to the power of zero equal to 1?
Dividing aⁿ by itself gives 1, and by the quotient rule it also gives aⁿ⁻ⁿ = a⁰. So a⁰ must be 1 for any non-zero a. The special case 0⁰ is usually taken as 1 in algebra but is left undefined in calculus.
How do I find the exponent if I know the result?
Use logarithms: if aˣ = c then x = log(c) ÷ log(a). For 3ˣ = 243, x = log 243 ÷ log 3 = 5. Choose “Find the exponent”, or use the Logarithm Calculator for any base.
How large a power can it calculate?
Exact answers are shown in full up to 100,000 digits. Bigger results — such as 2 to the power one million, which has 301,030 digits — are given as a digit count and in scientific notation, which the Scientific Notation Converter can also help you read.