🔢 Exponent Calculator
Raise any number to any power, including negative, fractional and very large exponents. The laws of exponents that govern all of it are set out underneath.
Powers, and the rules that make negative and fractional exponents obvious
An exponent counts how many copies of the base are multiplied together, which works perfectly while the exponent is a positive whole number. Two to the tenth is ten twos multiplied, 1,024. The question that makes exponents genuinely useful is what happens when the exponent is zero, negative, or a fraction — and the answers are not arbitrary conventions but the only values that keep one rule intact.
bm × bn = bm+n
Everything follows from insisting that this holds in all cases. If b³ × b⁰ must equal b³, then b⁰ has to be 1. If b³ × b⁻³ must equal b⁰ = 1, then b⁻³ has to be 1/b³. If b½ × b½ must equal b¹, then b½ has to be the square root of b.
The complete set of rules
| Rule | Form | Example |
|---|---|---|
| Product | bm × bn = bm+n | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | bm / bn = bm−n | 2⁵ / 2² = 2³ = 8 |
| Power of a power | (bm)n = bmn | (2³)² = 2⁶ = 64 |
| Zero | b0 = 1 | 7⁰ = 1 |
| Negative | b−n = 1 / bn | 2⁻³ = 0.125 |
| Fractional | b1/n = n√b | 81/3 = 2 |
| Mixed fraction | bm/n = (n√b)m | 82/3 = 4 |
The last row is the one worth practising, because it can be read two ways that give the same answer: take the cube root of 8 and square it, or square 8 to get 64 and take its cube root. Rooting first keeps the numbers small, which is why it is the better order by hand.
The minus sign trap
Exponents bind more tightly than multiplication, and more tightly than a leading minus sign. So −3² means −(3²) = −9, while (−3)² = 9. This is not a quirk of any particular calculator — it is the standard order of operations, and it accounts for a large share of the wrong answers people get from a calculator they have typed into correctly.
A negative base with a fractional exponent usually has no real answer, because it asks for an even root of a negative number. (−8)1/3 is −2 and perfectly real, but (−8)1/2 is not a real number at all. Odd roots of negatives exist; even roots do not.
How fast powers grow
| n | 2n | In words |
|---|---|---|
| 10 | 1,024 | A thousand |
| 20 | 1,048,576 | A million |
| 30 | 1,073,741,824 | A billion |
| 40 | 1,099,511,627,776 | A trillion |
| 64 | 1.84 × 1019 | Grains on the last chessboard square |
Every ten doublings multiplies by roughly a thousand, which is the single most useful fact about powers of two in computing: 2¹⁰ is a kilobyte's worth, 2²⁰ a megabyte's, 2³⁰ a gigabyte's. The number base converter shows the same structure from the binary side.
Scientific notation is just exponents
Writing 6.02 × 10²³ rather than a number with 24 digits is the same mechanism applied to tidiness. Multiplying such numbers means multiplying the fronts and adding the exponents; dividing means subtracting them. A double-precision floating point number, which is what this calculator and your browser use, tops out around 1.8 × 10³⁰⁸, which is why extreme powers return infinity rather than a value.
Underflow is the mirror image: numbers smaller than about 5 × 10⁻³⁴⁵ collapse to zero. Both limits matter when chaining powers, because an intermediate result can overflow even when the final answer would have been representable. For logarithms, the inverse operation, see the log calculator.
A worked example
A culture doubles every 20 minutes. After 8 hours there have been 24 doublings, so the population is multiplied by 2²⁴ = 16,777,216. Starting from 500 cells gives 8.39 × 10⁹ cells. To find how long to reach a billion-fold increase, solve 2ⁿ = 10⁹: n = log₂(10⁹) ≈ 29.9 doublings, just under ten hours. Growth problems posed as exponents are solved as logarithms, which is the practical reason the two tools sit side by side.
If this is one step in a larger decision, the root calculator is usually the next question, and the log calculator is worth a look while you are here.
Frequently Asked Questions (FAQs)
Why is anything to the power of zero equal to one?
Because the rule that b to the m times b to the n equals b to the m plus n has to keep working. If b cubed times b to the zero must equal b cubed, b to the zero can only be 1. The one exception is zero to the power of zero, which is left undefined in most contexts.
What does a negative exponent mean?
The reciprocal: b to the minus n is one over b to the n. So 2 to the minus 3 is one eighth, or 0.125. It does not make the result negative — it makes it a fraction.
What does a fractional exponent mean?
A root. b to the one over n is the nth root of b, so 8 to the one third is 2. A fraction like two thirds means both: take the cube root and then square it, which for 8 gives 4.
Is minus 3 squared equal to 9 or minus 9?
Written as minus 3 squared it is minus 9, because the exponent applies to the 3 before the minus sign is applied. Only (minus 3) squared, with the brackets, gives 9. This is standard order of operations, not a calculator quirk.
Can you raise a negative number to a fractional power?
Only when the root is odd. The cube root of minus 8 is minus 2, which is fine, but the square root of minus 8 is not a real number, so a negative base with an even root has no real answer.
How do I multiply numbers in scientific notation?
Multiply the leading numbers and add the exponents. So 3 times 10 to the 4 multiplied by 2 times 10 to the 5 is 6 times 10 to the 9. For division, divide the leading numbers and subtract the exponents.
Why does my calculator return infinity?
Because the result exceeded what double-precision floating point can hold, roughly 1.8 times 10 to the 308. Very small results hit the opposite limit and collapse to zero below about 5 times 10 to the minus 345.