What an exponent is
Write an and you are saying "multiply n copies of a together". So 23 is 2 × 2 × 2 = 8, and 210 is 1,024. The base is the number being repeated; the exponent counts the repetitions.
That definition only makes obvious sense when the exponent is a positive whole number — you cannot multiply a number by itself −3 times, or half a time. Everything else is defined by extension, and the extension is not arbitrary. There is exactly one way to define a0, a−n and a1/n that keeps the basic law am · an = am+n true, and those definitions are the ones everyone uses.
Work it out for yourself. If the law is to hold, then a3 · a0 must equal a3, so a0 has to be 1 for any non-zero a. Likewise a3 · a−3 = a0 = 1, so a−3 must be 1/a3. And a1/2 · a1/2 = a1 = a, so a1/2 is whatever number squares to a — the square root. None of these are conventions in the arbitrary sense; each is forced.
Exponents matter because they describe growth that feeds on itself: compound interest, populations, radioactive decay, algorithm run times, signal strength. A quantity growing by a fixed percentage each period is a power, and the exponent is the number of periods.
The laws of exponents, and why each one holds
Six rules cover essentially all exponent manipulation, and each is just bookkeeping on the count of factors.
Product rule: am · an = am+n. Three copies times four copies is seven copies. This is the rule from which the others follow.
Quotient rule: am ÷ an = am−n. Dividing cancels factors, so you subtract the counts. Notice what happens when n exceeds m: you get a negative exponent, which is the reciprocal.
Power of a power: (am)n = am·n. A group of m factors, taken n times over, is m × n factors.
Power of a product: (a·b)n = an·bn. Multiplication commutes, so you can sort the factors into two piles. There is no matching rule for sums: (a + b)2 is not a2 + b2, and treating it as though it were is the most common error in all of algebra.
Zero exponent: a0 = 1 for a ≠ 0. Forced by the quotient rule, since an ÷ an is both 1 and a0.
Fractional exponent: am/n = n√(am). The denominator is the root index and the numerator is the power. So 160.25 is the fourth root of 16, which is 2, and 82/3 is the cube root of 8 squared, which is 4.
Two special cases are worth naming. A negative base with a whole exponent is positive when the exponent is even and negative when it is odd, because the minus signs pair off. Be careful with notation: −24 means −(24) = −16, while (−2)4 = 16. A negative base with a fractional exponent is real only when the root index is odd. The cube root of −8 is −2, and this calculator returns it; the square root of −8 is not a real number and it returns nothing.
Worked example: 2¹⁰, 5⁻³ and 16^0.25 by hand
210. Start at 2 and double nine times: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. Ten factors of 2 take nine multiplications, and this is the sequence programmers know by heart. Shortcut: 210 = (25)2 = 322 = 1,024 — four multiplications to build 32 and one to square it, so the power-of-a-power rule turns nine multiplications into five.
5−3. Deal with the sign of the exponent last. First 53 = 5 × 5 × 5 = 125. Then the negative exponent means take the reciprocal: 1 ÷ 125 = 0.008. A common slip is to answer −125; a negative exponent never makes the result negative, it makes it a reciprocal. The sign of a power comes only from the sign of the base.
160.25. Write the decimal as a fraction: 0.25 = 1/4. A denominator of 4 means the fourth root, so you want the number that, multiplied by itself four times, gives 16. Try 2: 2 × 2 × 2 × 2 = 16. So the answer is 2. Check with the power-of-a-power rule: (161/4)4 = 161 = 16, and 24 = 16 agrees.
Putting them together. Evaluate 210 ÷ 24 × 2−3. Rather than computing three powers, add and subtract the exponents: 10 − 4 + (−3) = 3, so the answer is 23 = 8. Confirm the long way: 1,024 ÷ 16 = 64, and 64 × 0.125 = 8. The exponent arithmetic is a great deal less work, and that saving is the entire practical point of the laws.
Reading the result, and when to switch to logarithms
Powers get large fast, and the scientific-notation output is there because the plain decimal stops being readable long before the calculator stops being accurate. 264 is about 1.845 × 1019; the twenty-digit form tells you almost nothing that the mantissa and the exponent do not.
Beyond roughly 10308 the result exceeds what double-precision arithmetic can hold and the calculator reports nothing rather than a wrong number. When you are near that ceiling, stop computing the power and start computing its logarithm: log10(an) = n · log10(a), which stays comfortably small no matter how large the power. That is how you compare 3500 with 7300 without evaluating either — 500 log 3 = 238.6 against 300 log 7 = 253.5, so the second is larger. The logarithm calculator does this directly.
Watch precision at the other end too. Results below about 10−300 lose digits, and fractional powers of large bases carry small rounding errors because they are evaluated through exponentials and logarithms internally rather than by exact multiplication. If 160.25 returns 2.0000000000000004 in some tool, that is floating-point representation, not a mathematical claim.
Finally, be careful when comparing a power against a multiple. Exponential growth overtakes any fixed multiplier eventually, but "eventually" can be a long way out: 2n stays below 100n up to n = 9 (512 against 900), and only from n = 10 does the power take the lead (1,024 against 1,000). Check the crossover rather than assuming it.
Powers of common bases
| n | 2n | 3n | 5n | 10n |
|---|---|---|---|---|
| 1 | 2 | 3 | 5 | 10 |
| 2 | 4 | 9 | 25 | 100 |
| 3 | 8 | 27 | 125 | 1,000 |
| 4 | 16 | 81 | 625 | 10,000 |
| 5 | 32 | 243 | 3,125 | 100,000 |
| 6 | 64 | 729 | 15,625 | 1,000,000 |
| 7 | 128 | 2,187 | 78,125 | 10,000,000 |
| 8 | 256 | 6,561 | 390,625 | 100,000,000 |
| 9 | 512 | 19,683 | 1,953,125 | 1,000,000,000 |
| 10 | 1,024 | 59,049 | 9,765,625 | 10,000,000,000 |
The 10 column is why scientific notation uses base 10: the exponent counts the zeros. The 2 column is why memory sizes come in 256s and 1,024s rather than round decimal numbers.
Errors that survive a second look
- Reading a negative exponent as a negative answer. 5−3 is 0.008, not −125. The exponent's sign controls reciprocation; only the base's sign controls the answer's sign.
- Distributing a power over a sum. (a + b)2 = a2 + 2ab + b2, not a2 + b2. The middle term is not optional, and dropping it is the most expensive habit in elementary algebra.
- Mixing up −24 and (−2)4. The first is −16, the second is 16. Without brackets the exponent binds tighter than the minus sign.
- Multiplying the exponents when you should add. a3 · a4 = a7, while (a3)4 = a12. Multiplying powers adds; raising a power multiplies.
- Taking an even root of a negative number. (−8)1/3 = −2 is fine; (−8)1/2 has no real value. Only odd root indices survive a negative base.
- Rounding the base before raising it. Small errors compound with the exponent. Rounding 1.045 to 1.05 before raising to the 30th power overstates the result by about 15%.
Related tools
The inverse question — "to what power must I raise this base to get that number" — is a logarithm, and the logarithm calculator answers it in any base. Powers and logarithms undo each other exactly, so anything awkward in one is usually easy in the other.
When the exponent is one half, you are taking a square root, and the square root calculator gives the principal root along with the perfect-square check and the nearest squares either side. Large results are easier to read and compare in scientific notation, which the scientific notation calculator converts to and from, including engineering notation with exponents in multiples of three.
Repeated percentage growth is a power in disguise: a value rising 10% a year for n years is multiplied by 1.10n. The percentage increase and decrease calculator compounds that directly, and finding the rate behind a known total is a fractional power of the overall multiplier.
In number theory, exponents record how many times each prime divides a number: 360 = 23 × 32 × 5. The prime factorization calculator produces that form, and the divisor count follows from adding one to each exponent and multiplying. For counting problems where powers appear as arrangements with repetition, see the permutation calculator and the factorial calculator.