An exponent tells you how many times to multiply a number by itself. That one idea, plus three shortcuts, handles almost every power you will meet by hand, from small whole powers to negative and fractional ones.
How it works
For a positive whole exponent n, a^n means a multiplied by itself n times. Any non-zero number to the power 0 is 1. A negative exponent means the reciprocal: a^-n = 1 / a^n. A fractional exponent means a root: a^(1/n) is the nth root of a, so a^(1/2) is the square root and a^(1/3) is the cube root. When bases match, you can add exponents when multiplying and subtract them when dividing.
Step by step
- Write the base and count how many copies the exponent demands.
- Multiply step by step, writing each running total, so an error stays local.
- For a zero exponent, answer 1 at once, unless the base is 0, which is undefined as 0^0.
- For a negative exponent, compute the positive power first, then take the reciprocal.
- For a fractional exponent, find the root first, then apply any whole power left.
- Check by reversing: take the root of your answer and confirm you return to the base.
Worked example
3^4 = 3 x 3 x 3 x 3. Step by step: 3 x 3 = 9, 9 x 3 = 27, 27 x 3 = 81. 2^-3 = 1 / 2^3 = 1 / 8 = 0.125. 64^(1/3) = 4, because 4 x 4 x 4 = 64. 5^(3/2) = (square root of 5)^3, or the square root of 125, about 11.18. A decimal example: 1.5^3 = 1.5 x 1.5 x 1.5 = 3.375.
Common mistakes
- Multiplying base by exponent instead of multiplying base by itself exponent times. 3^4 is 81, not 12.
- Dropping the minus sign on a negative exponent and answering with the positive power.
- Applying the exponent to only part of a bracketed product. (2 x 5)^3 is 1000, not 2 x 125.
- Confusing -2^2, which is -4, with (-2)^2, which is 4. Brackets change the sign.
Try it with the free tool
You can follow every step above by hand, but the free tool below does the same calculation instantly. Try your own numbers in it, then check one result by hand so you trust the method as well as the answer. Open the full tool here: open the tool.
Checklist before you finish
- Write down every input number first, with its unit, before you calculate anything.
- Work through the steps in order, and do not skip the unit conversions.
- Check the answer against the worked example above: is it the right size, and does the direction make sense?
- Round only at the end, and keep the unrounded figure for any follow-up calculation.
Practice with your own numbers
Pick three cases of your own and work each one by hand. First, try a simple round-number case you can check mentally. Second, try a realistic case with awkward numbers, the kind you will actually meet. Third, try an edge case, such as a very small or very large input, and confirm the answer still behaves sensibly. If any case gives a surprising result, re-check the units and the formula step by step before you trust it.
If your answer looks wrong
Most errors come from a small set of causes. Check that you used the same units throughout, that you divided where the formula divides and multiplied where it multiplies, and that you applied any percentage as a decimal. Check the order of operations: powers and brackets come before multiplication and division, which come before addition and subtraction. Finally, test the method on the worked example in this guide. If your method reproduces the example answer, the method is right and the inputs deserve a second look.
Quick tips
For large exponents, split them: 2^10 = 2^5 x 2^5 = 32 x 32 = 1024. This keeps the numbers small and checkable at each step.