Exponent and power calculator
Raise any number to an integer, decimal or negative exponent.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Raise any number to an integer, decimal or negative exponent.
- Formula used
aⁿ = a × a × … × a (n times)- Example
- 2¹⁰ = 1024
Exponent calculator
Raising to a power means multiplying the base by itself as many times as the exponent says, but the definition extends to negative and fractional exponents.
Decimal and negative bases are accepted, with the single condition that a negative base requires a whole exponent.
How it works
A negative exponent is the reciprocal of the positive power. A fractional exponent is a root: raising to 1/3 is the same as taking the cube root.
Formula
aⁿ = a × a × … × a (n times)
a⁻ⁿ = 1 / aⁿ
a^(m/n) = ⁿ√(aᵐ)
a⁰ = 1 for every a ≠ 0
Worked example
2¹⁰ = 1024
2⁻³ = 1 / 2³ = 0.125
8^(1/3) = ³√8 = 2
Explanation
Why any number to the power of zero is one
It is not an arbitrary convention. Dividing powers of the same base subtracts the exponents: two cubed divided by two cubed is two to the zero, and it is also one because any number divided by itself is one. The definition is chosen precisely so that the rules keep working without exceptions.
Negative exponents are reciprocals
Two to the minus three is one over two cubed, that is an eighth. It follows from the same rule of subtracting exponents when dividing, extended past zero. It does not mean the result is negative: it means it is inverted.
Exponential growth defeats intuition
Doubling something thirty times starting from one gives over a billion. It is why compound interest, epidemics and storage capacity always surprise: human intuition reasons linearly while reality advances by multiplying.
Order matters without brackets
Two to the three to the two is 512 if resolved top down, and 64 if the base is grouped with the first exponent. The mathematical convention is the former, but in a spreadsheet it is wiser to use brackets than to trust it.
Frequently asked questions
What is any number to the power of zero?
One, as long as the base is not zero. The expression 0⁰ is left undefined.
Why can a negative base not take a decimal exponent?
Because (−8)^0.5 would be the square root of a negative number, which does not exist among the reals.
Why does a negative exponent not give a negative result?
Because it indicates the reciprocal, not the sign. Two to the minus three is an eighth, a positive number smaller than one.
How do you resolve a power of a power?
The mathematical convention resolves from the top down. To avoid ambiguity use brackets, especially in spreadsheets and programming languages.
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