Exponent Calculator

Exponents are repeated multiplication, compressed

Writing 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 is tedious and easy to miscount, so mathematics compresses it into a power: a small raised exponent counts how many copies of the base are multiplied together. In 2¹⁰ the base is 2, the exponent is 10, and the answer is 1024 — a number so familiar in computing, where a kilobyte was long defined around it, that it is worth remembering outright. This calculator evaluates any base to any exponent, and whenever both inputs are whole numbers with a non-negative exponent it also works the answer out with exact big-integer arithmetic, so the decimal string it shows is correct digit for digit, not a rounded approximation.

That exactness matters more than it first appears. Ordinary computer arithmetic carries about 15 or 16 reliable decimal digits; beyond that, a power such as 2¹⁰⁰ would normally be printed in scientific notation with the later digits lost. Exact integer arithmetic keeps every one of its 31 digits. For everyday school and finance work the ordinary result is plenty, but for checking patterns — the last digit of a huge power, the exact size of a chessboard-and-rice doubling problem — the exact string is the whole point.

The rules that make powers predictable

Growth, doubling and why exponents surprise people

Linear growth adds the same amount each step; exponential growth multiplies by the same amount each step, and multiplication outruns addition startlingly fast. Paper folded in half (in the idealised maths, not the physical desk) doubles in thickness each fold: after 10 folds it is about a thousand times thicker, after 42 folds it would pass the Moon. Compound interest, population growth, viral spread and radioactive decay — decaying by a half-life is just a power with a fraction as its base — are all powers wearing different clothes. When an exponent answer looks absurdly large or small, that is usually the mathematics being honest about how fast repeated multiplication really is.

Special cases and honest limits

A few inputs deserve a warning rather than a number. Zero to the power zero is undefined in most school contexts and is left as a convention, not a fact. A negative base with a fractional exponent, such as (−8) to the power one third in real arithmetic done naively, can produce a non-real result that ordinary decimal arithmetic reports as NaN; the honest response is a note, not a pretend answer. Very large exponents overflow ordinary decimals into infinity — the exact integer string is still shown for whole-number powers of moderate size, which is exactly the case where you most want every digit. If you need the inverse question — to what power must I raise this base to reach that number — that is a logarithm, covered by its own calculator in this section.

Worked example

2 to the power 10: write 2 multiplied by itself 10 times and the result is 1024. By contrast 10 to the power 3 is 1000, and 5 to the power −2 is 1 ÷ 25 = 0.04.

How to use this calculator

1. Enter the base — the number being multiplied.
2. Enter the exponent — how many copies, or a negative or fractional power.
3. Read the result in the green panel; whole-number powers also show the exact digit string.
4. Check the breakdown for the digit count and the ordinary decimal form.
5. Follow the working panel to see the power written out and the rule applied.

Powers of two and powers of ten

Two families of powers run modern life. Powers of ten give the metric system and scientific notation its skeleton: a million is 10⁶, a billion 10⁹, and moving the exponent by one moves the decimal point by one place. Powers of two give computing its skeleton: 2⁸ = 256 values in a byte, 2¹⁰ = 1024, 2¹⁶ = 65,536 and 2³² is about 4.3 billion — the address space that older internet protocols were built on. Learning to move fluently between the families, using the fact that 2¹⁰ is very close to 10³, is the estimation trick behind most back-of-envelope calculations in science and engineering.

Frequently asked questions