Exponent Calculator
Raises any base to any power, including negative and fractional exponents, and shows the working.
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Result
- 2^10
- 1024
- Written out
- 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
2 to the power of 10…
aⁿ = a × a × … × a (n times)
2 multiplied by itself 10 times
2^10 = 1024
Formula — written step by step
2 to the power of 10…
aⁿ = a × a × … × a (n times)
2¹ = 2, 2² = 4, 2³ = 8
Each step multiplies by another 2
2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
2¹⁰ = 1,024
a is the base — the number being multiplied — and n the exponent, which counts how many copies are multiplied together. A negative exponent means a reciprocal, and a fractional one means a root.
A worked example
A negative exponent
- Base
- 5
- Exponent
- −3
- A negative exponent inverts the base
- 5⁻³ = 1 ÷ 5³
- 5³ = 125
- 1 ÷ 125 = 0.008
5⁻³ = 0.008
A negative exponent never makes the result negative — it makes it a fraction. 5⁻³ is one over five cubed, which is small and positive. Only a negative base can produce a negative answer, and only when the exponent is odd.
How to use this calculator
- Enter the base — the number being raised to a power.
- Enter the exponent. Negative values are allowed and produce a reciprocal; fractional values produce a root.
- Read the result, with the multiplication written out in full for small whole exponents.
- For very large or very small results, use the scientific-notation row rather than counting zeros.
What an exponent does
An exponent is shorthand for repeated multiplication. Writing 2¹⁰ instead of ten twos multiplied together saves space, but more importantly it makes the structure visible: the exponent is a count, and the rules for combining exponents are really rules for adding and subtracting those counts.
That is why aᵐ × aⁿ = aᵐ⁺ⁿ. Multiplying three twos by four twos gives seven twos. Once you see the exponent as a tally of how many copies are present, every other rule follows without memorisation.
Negative and fractional exponents
A negative exponent is not a negative number — it is a reciprocal. 2⁻³ means one over 2³, or 0.125. The rule comes straight from division: 2³ ÷ 2⁶ is 2⁻³ by the quotient rule, and is also 8 ÷ 64 = 0.125 by arithmetic. Both must agree, so the negative exponent has to mean a reciprocal.
A fractional exponent is a root. 8^(1/3) is the cube root of 8, which is 2. This follows from the power-of-a-power rule: (8^(1/3))³ = 8^1 = 8, so 8^(1/3) must be the number that gives 8 when cubed.
Why negative bases are restricted
A negative base raised to a whole-number power is straightforward: the sign alternates, negative for odd exponents and positive for even ones, because pairs of negatives cancel. (−2)³ is −8 and (−2)⁴ is 16.
Fractional exponents on a negative base are a different matter. (−4)^(1/2) asks for a number that squares to −4, and no real number does. The calculator reports this rather than returning a value, because the answer exists only in the complex numbers, which the page has no way to display honestly.
The exponent rules
| Rule | Form | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1 ÷ aⁿ | 2⁻³ = 0.125 |
| Fractional exponent | a^(1/n) = ⁿ√a | 8^(1/3) = 2 |
Every rule follows from the definition. a⁰ = 1 because aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself is one.
Frequently asked questions
Why is any number to the power of zero equal to one?
Because of the quotient rule. aⁿ ÷ aⁿ is a⁰ by the rule, and is also 1 because anything divided by itself is one. Both must hold, so a⁰ = 1. The single exception is 0⁰, which is left undefined.
Is 2¹⁰ the same as 10²?
No, and the difference is large. 2¹⁰ is 1,024 while 10² is 100. Exponentiation is not commutative — the base and the exponent do very different jobs.
What does a fractional exponent like 2^1.5 mean?
It combines a power and a root: 2^1.5 is 2^(3/2), the square root of 2 cubed, which is about 2.828.
Why is (−2)⁴ positive but −2⁴ negative?
Because of what the exponent attaches to. In (−2)⁴ the whole negative number is raised to the fourth power, giving 16. In −2⁴ only the 2 is raised, and the minus sign is applied afterwards, giving −16.