Exponents compress repeated multiplication into a single elegant notation. Once you internalise the seven rules below, simplifying expressions like $\frac{x^5 y^{-2}}{x^{-3} y^4}$ becomes a 30-second exercise. This page is the cheat sheet you can keep open during homework.

Why exponents matter

The exponent rules are not arbitrary — they all follow from the definition $a^n = \underbrace{a \cdot a \cdots a}_{n \text{ copies}}$ . Once you see why each rule works, you stop memorising and start deriving on demand.

The seven core laws

#	Law	Example
1	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^4 = x^7$
2	$a^m / a^n = a^{m-n}$	$x^7 / x^2 = x^5$
3	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
4	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
5	$(a/b)^n = a^n / b^n$	$(x/y)^4 = x^4/y^4$
6	$a^{-n} = 1/a^n$	$x^{-3} = 1/x^3$
7	$a^{m/n} = \sqrt[n]{a^m}$	$8^{2/3} = (\sqrt[3]{8})^2 = 4$

Plus the two definitional cases: $a^0 = 1$ for any $a \ne 0$ , and $a^1 = a$ .

Worked example: combining rules

Simplify $\frac{(2x^3)^2 \cdot x^{-4}}{4x^{-1}}$ .

Apply rule 4 to the bracket: $(2x^3)^2 = 4x^6$ .
Substitute: $\frac{4x^6 \cdot x^{-4}}{4x^{-1}}$ .
Cancel the 4s: $\frac{x^6 \cdot x^{-4}}{x^{-1}}$ .
Combine numerator with rule 1: $\frac{x^2}{x^{-1}}$ .
Apply rule 2: $x^{2 - (-1)} = x^3$ .

The whole simplification is just bookkeeping — the rules carry you.

Negative and fractional exponents intuition

A negative exponent does not mean "negative number"; it means reciprocal. So $5^{-2} = 1/25$ , not $-25$ .

A fractional exponent $a^{p/q}$ is root-then-power (or power-then-root, same answer). The denominator picks the root, the numerator picks the power: $32^{3/5} = (\sqrt[5]{32})^3 = 2^3 = 8$ .

Common mistakes

$(a + b)^n \ne a^n + b^n$ — exponents don't distribute over addition. $(2 + 3)^2 = 25$ , not $4 + 9$ .
$a^{-n} \ne -a^n$ — negative exponent is reciprocal, not negation.
$0^0$ is conventionally $1$ in algebra and combinatorics, but undefined in some analysis contexts. Beware when in doubt.

Try with the AI Exponent Solver

Paste any expression into the Exponent / Simplify Solver and you'll get a step-by-step simplification using exactly the rules above.

Exponent Rules Explained: Every Law With Worked Examples

A clean walkthrough of all exponent laws — products, quotients, powers of powers, negative and fractional exponents — with side-by-side worked examples.