Raise any number to any power — positive, negative, fractional, decimal or zero — with the working, the exponent laws and scientific notation.
An exponent tells you how many times to multiply a number by itself. In 2⁵, the base is 2 and the exponent is 5, so 2 × 2 × 2 × 2 × 2 = 32. A negative exponent means a reciprocal, so 2⁻³ = 1/8 = 0.125. A fractional exponent means a root, so 8^(1/3) = 2. Enter a base and an exponent below to see the result and every step.
Both may be decimals or negative. Fractions work too — enter 0.3333333333 or 1/3 for a cube root.
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An exponent, also called a power or index, is shorthand for repeated multiplication. Writing 2⁵ saves writing 2 × 2 × 2 × 2 × 2. The number being multiplied is the base, and the small raised number is the exponent.
Read 2⁵ as "two to the power of five", or "two to the fifth". Two special cases have their own names: b² is "b squared" and b³ is "b cubed", from the area of a square and the volume of a cube.
Enter the base and the exponent, then press Calculate or the Enter key. The display above the inputs shows your expression as it will be read. Results include the value in plain and scientific notation, the inverse operation, and a step-by-step explanation that changes depending on whether your exponent is positive, negative, fractional or zero. Tap any example button to load a preset, or use Share link to send someone the exact calculation.
| Law | Rule | Example | Why |
|---|---|---|---|
| Product | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 | Three 2s times four 2s is seven 2s |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 | Two of the 2s cancel out |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 | Three 2s, taken twice over |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2×3)² = 4 × 9 = 36 | Each factor gets the power |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (3/2)² = 9/4 = 2.25 | Top and bottom each get the power |
| Zero exponent | b⁰ = 1 | 7⁰ = 1 | Follows from bⁿ ÷ bⁿ = 1 |
| Negative exponent | b⁻ⁿ = 1/bⁿ | 2⁻³ = 1/8 = 0.125 | Continues the halving pattern below b⁰ |
| Fractional exponent | b^(1/n) = ⁿ√b | 8^(1/3) = 2 | Because (b^(1/n))ⁿ = b¹ |
This surprises people, but it follows from the quotient law. Since bⁿ ÷ bⁿ must equal 1, and the law says it also equals bⁿ⁻ⁿ = b⁰, the two must agree: b⁰ = 1.
The pattern makes it visible. Look at the powers of 2 going down: 2³ = 8, 2² = 4, 2¹ = 2. Each step halves the result, so the next one is 2⁰ = 1, then 2⁻¹ = 0.5, then 2⁻² = 0.25. Nothing special happens at zero — the pattern simply continues.
The one contested case is 0⁰. Different branches of mathematics treat it differently, but in algebra and computing it is almost always defined as 1, which is what this calculator returns.
A negative exponent means take the reciprocal — flip it over. It does not make the answer negative.
So 2⁻³ = 1/2³ = 1/8 = 0.125, and 10⁻² = 1/100 = 0.01. This is the most common exponent mistake: −3 in the exponent tells you to invert, while a minus sign in front of the whole expression would make it negative. Compare 2⁻³ = 0.125 with −2³ = −8.
A fractional exponent is a root. The denominator gives which root, and the numerator gives the power:
So 8^(1/3) = ³√8 = 2, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Decimal exponents work the same way — 9^0.5 is just 9^(1/2) = 3.
Negative bases need care here. The cube root of −8 is −2, because (−2)³ = −8, so (−8)^(1/3) has a perfectly good real answer. But the square root of −4 does not exist in real numbers, because no real number squared gives a negative. The rule is that odd roots of negatives are fine and even roots are not, and this calculator applies exactly that test rather than returning an error for every negative base.
| n | 2ⁿ | 3ⁿ | 5ⁿ | 10ⁿ |
|---|---|---|---|---|
| −2 | 0.25 | 0.111111 | 0.04 | 0.01 |
| −1 | 0.5 | 0.333333 | 0.2 | 0.1 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 2 | 3 | 5 | 10 |
| 2 | 4 | 9 | 25 | 100 |
| 3 | 8 | 27 | 125 | 1,000 |
| 4 | 16 | 81 | 625 | 10,000 |
| 5 | 32 | 243 | 3,125 | 100,000 |
| 6 | 64 | 729 | 15,625 | 1,000,000 |
| 8 | 256 | 6,561 | 390,625 | 100,000,000 |
| 10 | 1,024 | 59,049 | 9,765,625 | 10,000,000,000 |
| Root form | Exponent form | Example |
|---|---|---|
| √b | b^(1/2) | √16 = 16^0.5 = 4 |
| ³√b | b^(1/3) | ³√27 = 27^(1/3) = 3 |
| ⁴√b | b^(1/4) | ⁴√16 = 16^0.25 = 2 |
| ⁿ√(bᵐ) | b^(m/n) | 16^(3/4) = 8 |
| 1/√b | b^(−1/2) | 16^(−0.5) = 0.25 |
Writing roots as exponents means the same set of laws covers both, which is why algebra prefers the exponent form.
Exponents and logarithms are inverse operations — two ways of describing the same relationship. An exponent starts with the base and the power and finds the result; a logarithm starts with the base and the result and finds the power.
| Feature | Exponents | Logarithms |
|---|---|---|
| Question asked | What is 2 to the power 10? | 2 to what power gives 1024? |
| What it does | Raises a number to a power | Finds the exponent |
| Direction | Forward operation | Inverse operation |
| Written | 2¹⁰ = 1024 | log₂(1024) = 10 |
| Effect on size | Grows numbers rapidly | Compresses large ranges |
| Turns… | Addition into multiplication | Multiplication into addition |
| Typical use | Compound interest, growth models | pH, decibels, algorithm analysis |
bˣ = y ⟷ log_b(y) = xlog_b(bˣ) = x and b^log_b(y) = y
If you have the power and want the result, you are on the right page. If you have the result and want the power, use the Log Calculator instead — it handles any base, antilogs and the change of base rule.
A few sequences come up constantly, and recognising them saves time in exams and in code.
The foundation of computing, because each bit doubles the possibilities. Worth memorising up to 2¹⁰.
| 2¹ | 2² | 2³ | 2⁴ | 2⁵ | 2⁶ | 2⁷ | 2⁸ | 2⁹ | 2¹⁰ |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1,024 |
2⁸ = 256 is why a byte holds 256 values, and 2¹⁰ = 1024 is why a kilobyte was originally 1024 bytes. Handy shortcut: 2¹⁰ ≈ 1 thousand, 2²⁰ ≈ 1 million, 2³⁰ ≈ 1 billion.
The exponent simply counts the zeros, which is what makes scientific notation work.
| Power | Value | Name | Prefix |
|---|---|---|---|
| 10⁻³ | 0.001 | One thousandth | milli |
| 10⁻² | 0.01 | One hundredth | centi |
| 10⁰ | 1 | One | — |
| 10³ | 1,000 | One thousand | kilo |
| 10⁶ | 1,000,000 | One million | mega |
| 10⁹ | 1,000,000,000 | One billion | giga |
| 10¹² | 1,000,000,000,000 | One trillion | tera |
A perfect square is a whole number times itself. Recognising them makes simplifying roots much faster.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Perfect squares always end in 0, 1, 4, 5, 6 or 9 — so any number ending in 2, 3, 7 or 8 cannot be one.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1,000 |
Unlike squares, cubes keep the sign of the original number — (−3)³ = −27 — which is exactly why odd roots of negative numbers exist.
Growth happens when the base is greater than 1. Compound interest at 5% multiplies by 1.05 every year, so after t years the factor is 1.05ᵗ. Growth like this starts slowly and then accelerates sharply — the reason compound interest is so powerful over long periods.
Decay happens when the base is between 0 and 1. A radioactive isotope halving each period follows 0.5ᵗ, which is the same as 2⁻ᵗ. The quantity falls quickly at first, then more and more slowly, approaching zero without ever reaching it.
Compound interest multiplies by the same factor each period, so growth is a power of the rate.
A = P(1 + r)ᵗSoftmax and activation functions use exponentials to turn scores into probabilities.
softmax = eˣ / ΣeˣMemory, addressing and data types are all powers of two, which is why sizes jump 256, 512, 1024.
2ⁿ bytesRadioactive decay halves over a fixed period, following an exponential curve.
N = N₀ · (½)^(t/h)Bacteria doubling each generation grow as powers of two, which is textbook exponential growth.
N = N₀ · 2ᵗAreas scale with the square of length and volumes with the cube, which governs scaling limits.
A ∝ L², V ∝ L³Very large and very small measurements are written as a power of ten to stay readable.
3.0 × 10⁸ m/sKey strength is measured in powers of two, so each extra bit doubles the work to break it.
2²⁵⁶ combinations1) 81 2) 100,000 3) 1 4) 0.25 5) 5
1) 2⁸ = 256 2) 3⁶ = 729 3) 8 4) −3 5) 0.125
What is an exponent?
An exponent tells you how many times to multiply a number by itself. In 2 to the power 5, the base is 2 and the exponent is 5, giving 2 times 2 times 2 times 2 times 2, which is 32.
What is the difference between a base and an exponent?
The base is the number being multiplied and the exponent is the small raised number saying how many times. In 3 to the power 4, the base is 3 and the exponent is 4.
How do you calculate exponents?
Multiply the base by itself as many times as the exponent says. For decimal or fractional exponents, use a calculator, since they represent roots rather than simple repeated multiplication.
What are the exponent rules?
Multiplying powers with the same base adds the exponents, dividing subtracts them, and raising a power to a power multiplies them. A zero exponent gives 1, a negative exponent gives a reciprocal, and a fractional exponent gives a root.
Why is any number to the power of zero equal to 1?
Because dividing a power by itself must give 1, and the quotient rule says that is the base to the power zero. The pattern of halving also continues naturally through zero.
What is zero to the power zero?
It is contested. In algebra, combinatorics and most programming languages it is defined as 1, which is what this calculator returns, though some areas of analysis leave it undefined.
What is a negative exponent?
A negative exponent means take the reciprocal. So 2 to the power minus 3 equals 1 divided by 2 cubed, which is 0.125. The answer is positive, not negative.
Does a negative exponent make the answer negative?
No. It makes the answer a fraction of 1. Only a negative base with an odd exponent gives a negative result, such as minus 2 cubed equalling minus 8.
What is a fractional exponent?
A fractional exponent is a root. The denominator says which root and the numerator says which power, so 16 to the power three quarters is the fourth root of 16 cubed, which is 8.
How do decimal exponents work?
They work exactly like fractions. A power of 0.5 is a square root and a power of 0.25 is a fourth root, because those decimals are one half and one quarter.
Can you raise a negative number to a fractional power?
Sometimes. Odd roots work, so the cube root of minus 8 is minus 2. Even roots do not, because no real number squared gives a negative result.
What is the difference between minus 3 squared and minus 3 in brackets squared?
The power binds more tightly than the minus sign, so minus 3 squared is minus 9. With brackets, minus 3 in brackets squared is positive 9.
How do you multiply exponents?
When the bases match, add the exponents. So 2 cubed times 2 to the fourth equals 2 to the seventh, which is 128. If the bases differ, work each power out separately.
How do you divide exponents?
When the bases match, subtract the exponents. So 2 to the fifth divided by 2 squared equals 2 cubed, which is 8.
What is a power of a power?
Raising a power to another power multiplies the exponents. So 2 cubed, all raised to the power 2, equals 2 to the sixth, which is 64.
What is exponential growth?
Exponential growth happens when a quantity multiplies by the same factor each period, so the base is greater than 1. Compound interest and population growth both behave this way.
What is exponential decay?
Exponential decay happens when the base is between 0 and 1, so the quantity shrinks by the same proportion each period. Radioactive half-life is the classic example.
How are exponents used in finance?
Compound interest raises the growth factor to the power of the number of periods. One thousand dollars at five per cent for ten years becomes 1000 times 1.05 to the power 10, about 1628 dollars.
How are exponents used in computing?
Computer memory and data sizes are powers of two, which is why capacities jump from 256 to 512 to 1024. Encryption strength is also measured in powers of two.
What is scientific notation?
Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten, such as 3 times 10 to the power 8 for the speed of light in metres per second.
Is this Exponent Calculator free?
Yes. It is free with no sign-up, works on any device, and runs entirely in your browser so your entries stay private.