An exponent tells you how many times to multiply a number by itself: 2^10 means ten 2s multiplied together, which comes to 1,024. Five rules handle almost everything else. Multiplying powers with the same base adds the exponents; dividing subtracts them; a power raised to a power multiplies them; any nonzero number to the zero power equals 1; and a negative exponent means the reciprocal, so 5^-2 = 1/25 = 0.04. Fractional exponents are roots in disguise — a^(1/2) is the square root of a. This guide works through each rule with the arithmetic shown, and ends with the sign errors that catch nearly everyone once.
Start with the definition, because every rule is just the definition applied patiently. In a^n, a is the base and n is the exponent or power. 2^3 = 2 × 2 × 2 = 8. The exponent counts factors, nothing more mysterious. When a problem mixes rules, expand a small version by hand to check which rule applies — expanding 2^3 × 2^2 as (2×2×2) × (2×2) makes the product rule obvious before you trust it. For the numbers themselves, the exponent calculator handles bases and powers too large to expand, including the negative and fractional cases below.
The core exponent rules
The core rules, each with a one-line example. Product rule: a^m · a^n = a^(m+n) — for example 2^3 · 2^4 = 2^7 = 128, because three 2s times four 2s is seven 2s. Quotient rule: a^m ÷ a^n = a^(m-n) — 5^6 ÷ 5^2 = 5^4 = 625. Power of a power: (a^m)^n = a^(m·n) — (3^2)^3 = 3^6 = 729, since three groups of two 3s make six. Power of a product: (a·b)^n = a^n · b^n — (2·5)^3 = 10^3 = 1,000, and indeed 2^3 · 5^3 = 8 · 125 = 1,000. One condition matters throughout: the product and quotient rules need the same base. 2^3 · 3^2 has no shortcut — it is simply 8 · 9 = 72.
Powers of ten and scientific notation
Powers of ten deserve a detour, because they are the exponents you will meet most often. 10^2 = 100, 10^3 = 1,000, 10^6 = 1,000,000 — each step appends a zero, which is the entire logic of scientific notation: 4.2 × 10^6 simply means 4.2 million, with the exponent counting the places the decimal point moves. Negative powers of ten count places the other way: 10^-3 = 0.001. Metric prefixes are powers of ten in uniform — kilo is 10^3, mega is 10^6, milli is 10^-3, micro is 10^-6 — so converting 5 kilometers to meters (5 × 10^3 m) or 250 milligrams to grams (0.25 g, since a milligram is 10^-3 g) is the product and quotient rules wearing lab coats.
The zero exponent rule
The zero exponent rule confuses people because it feels like a trick, but it is bookkeeping, not magic: a^0 = 1 for any a except 0 itself. The quotient rule forces it — 2^3 ÷ 2^3 must equal both 1 (anything divided by itself) and 2^(3-3) = 2^0, so 2^0 has to be 1 for the rules to agree. The countdown pattern shows the same thing: 2^3 = 8, 2^2 = 4, 2^1 = 2, and each step down divides by 2, so 2^0 = 1. The special case 0^0 is genuinely contested — many school courses leave it undefined, while some advanced contexts define it as 1 — so follow your course's convention and do not let a calculator's answer settle a classroom question.
Negative exponents: the reciprocal rule
A negative exponent flips the number into a fraction: a^-n = 1 / a^n. Worked example: 5^-2 = 1 / 5^2 = 1/25 = 0.04. The countdown pattern from the zero rule continues smoothly through it: 2^0 = 1, 2^-1 = 1/2, 2^-2 = 1/4, 2^-3 = 1/8 — each step down still divides by 2. Negative exponents are how very small quantities get written compactly, which is why scientific notation uses them for measurements like 3 × 10^-4. One caution: the negative belongs to the exponent only. 2^-2 is positive 1/4, not negative — the sign of the result depends on the base and the parity of the power, not on the minus in the exponent.
Fractional exponents are roots in disguise
A fractional exponent is a root wearing a different coat: a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a^m — equivalently, the mth power of the nth root. Worked example: 27^(2/3). Take the cube root of 27 first: 27^(1/3) = 3, because 3 × 3 × 3 = 27. Then square it: 3^2 = 9. So 27^(2/3) = 9. Doing the root before the power keeps the numbers small; the other order works too — 27^2 = 729, and the cube root of 729 is 9 — but the arithmetic is heavier. The most-used cases are a^(1/2) (square root, so 9^(1/2) = 3) and a^(1/3) (cube root, so 8^(1/3) = 2).
Worked example: how fast 2^10 grows
To feel how fast powers grow, expand 2^10 one doubling at a time: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. Ten doublings multiply a quantity by just over a thousand; twenty doublings (2^20 = 1,048,576) multiply by about a million. This is why repeated doubling escapes intuition so quickly — a rumor, a bacteria colony or a chain letter does not grow by adding a steady amount but by multiplying by a steady factor, and multiplication compounds. It is also why the same exponent rules quietly run compound growth, where (1 + r)^n applies the power-of-a-product thinking to money over n periods. Whenever a process multiplies rather than adds, expect an exponent to be hiding in it.
Roots are the inverse operation, and thinking of them that way locks in the fractional rules. A square root undoes a square: since 5^2 = 25, √25 = 5, and 25^(1/2) = 5 is the same statement. A cube root undoes a cube: 4^3 = 64, so 64^(1/3) = 4. This gives you a free error check on any fractional power — raise your answer back to the denominator's power and see whether the original base returns. It also explains why even roots of negative numbers have no real answer: no real number squared gives a negative, so (-9)^(1/2) fails in the reals, while odd roots of negatives work fine — (-8)^(1/3) = -2, because (-2)^3 = -8.
Classic exponent mistakes
Now the classic errors, starting with the most tested one: -2^2 = -4 but (-2)^2 = 4. Without parentheses, the exponent binds to the 2 before the minus sign is applied, so -2^2 means -(2^2) = -4; with parentheses, the base is -2, and (-2) × (-2) = 4. Second error: adding powers by adding or multiplying bases. 2^3 + 2^3 is two copies of 8, which is 16 — and since 16 = 2^4, the sum equals 2^4, not 4^3 (which is 64) and not 4^6. Addition of powers has no exponent rule at all; only multiplication and division do. Third: applying the product rule across different bases, as in claiming 2^3 · 3^3 = 6^6 — the correct move is the power-of-a-product rule in reverse: 2^3 · 3^3 = (2·3)^3 = 6^3 = 216. Fourth: reading 5^-2 as -25 or -10 instead of 1/25. Each error is a rule applied one situation too far.
Where exponents show up in everyday life
Exponents surface in more everyday places than school suggests. Areas and volumes scale by powers: double a square's side and its area grows by 2^2 = 4, double a cube's edge and volume grows by 2^3 = 8 — which is why a slightly bigger storage box holds so much more. The same squaring relationship drives the vertex and graph work in how to find the vertex of a parabola, where the x^2 term shapes the curve. Measurement scales like decibels and pH are built on powers of ten, and the averages in mean, median and mode explained sit one step away from the squared deviations used in statistics. You do not need to love the notation to benefit from recognizing when something is being multiplied repeatedly.
A checking routine that catches slips
A reliable checking routine prevents most slips. First, identify the base and confirm whether parentheses make a negative number part of it. Second, name the operation — multiplying, dividing, raising to a power, or adding (in which case no exponent rule applies). Third, apply exactly one rule, then simplify the arithmetic. Fourth, sanity-check the size: a positive power of a number bigger than 1 must grow, a negative exponent must shrink below 1, and a fractional power of a large number lands between 1 and the number itself. The vertex form calculator and the mean median mode calculator both lean on the same habits of checking a squared or averaged result for plausibility before trusting it.
Calculator and spreadsheet notation causes its own crop of errors. The ^ key (or x^y on some models) is the power key, and the E in a result like 1.2E6 is not an error but scientific notation, meaning 1.2 × 10^6 = 1,200,000. Order of operations puts powers before multiplication and division, and before addition: 2 + 3^2 is 2 + 9 = 11, not 25. Parentheses outrank everything, which is the machinery behind the -2^2 trap — a calculator reads -2^2 exactly as written, as -(2^2) = -4, and only (-2)^2 returns 4. When a result looks absurd, re-enter it with explicit parentheses around the base before you doubt the rules.
Keep the five rules on one mental index card: same base multiplying adds powers, dividing subtracts them, a power of a power multiplies them, a^0 = 1, and a negative power means take the reciprocal — with fractional powers as roots for good measure. Every worked example in this guide, from 5^-2 = 0.04 to 27^(2/3) = 9, is those rules applied one step at a time with the arithmetic shown. When a base or exponent gets ugly — decimals, big powers, roots of roots — run it through the exponent calculator, then use the size checks above to confirm the answer behaves the way the rules say it should.