Formula

Exponent rules

Also written: laws of exponents · product rule for exponents · power of a power

a^m \times a^n = a^{m+n}

To multiply powers of the same base, add the exponents. To divide them, subtract. To raise a power to another power, multiply. Every one of these is counting factors, and all of them need the bases to match.

What each part means

$a$: the base — the number being multiplied, and the same in every term
$m, n$: the exponents, counting how many factors each power contributes

When to use it

Powers of the same base are multiplied, divided, or raised to another power. Not applicable when the bases differ — 2³ × 3⁴ has to be worked out separately.

The rules

a^m \times a^n = a^{m+n}

\frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn} \qquad (ab)^n = a^n b^n

Rule	Form	Because
product	$a^m a^n = a^{m+n}$	factors placed side by side
quotient	$a^m / a^n = a^{m-n}$	factors cancel in pairs
power of a power	$(a^m)^n = a^{mn}$	$n$ groups of $m$ factors
product in a power	$(ab)^n = a^n b^n$	the exponent reaches everything

Why they add rather than multiply

$2^3 \times 2^4$ is three $2$ s written next to four $2$ s, which is seven $2$ s:

(2 \times 2 \times 2)(2 \times 2 \times 2 \times 2) = 2^7

A power of a power is different, and that is where the multiplying comes from. $(2^3)^4$ is four copies of $2^3$ , so there are four groups of three factors:

3 \times 4 = 12 \quad\Rightarrow\quad (2^3)^4 = 2^{12}

Full worked examples are in the exponent rules lesson.

The bases have to match

2^3 \times 3^4 \quad\text{cannot be combined}

Counting needs identical things to count, so $2$ s and $3$ s stay apart. Work each power out and multiply the results: $8 \times 81 = 648$ .

They hold for zero and negative exponents too

The rules were built on counting factors, but they keep working once the exponents go past zero:

\frac{2^3}{2^5} = 2^{-2} = \frac{1}{4} \qquad \frac{2^5}{2^5} = 2^0 = 1

That consistency is the reason $a^0 = 1$ and $a^{-n} = \tfrac{1}{a^n}$ are defined the way they are. It is also what makes scientific notation arithmetic work — front numbers multiply while the exponents add.

Lessons that teach this

Exponent Rules: Multiplying, Dividing and Powers of Powers
The three core exponent rules and why each one comes down to counting factors: add the exponents to multiply, subtract to divide, multiply for a power of a power.
Negative and Zero Exponents: Why x⁰ = 1
Why any number to the power of zero equals 1, what a negative exponent means, and how to convert between negative powers and fractions without guessing.
Scientific Notation: Writing Very Large and Very Small Numbers
How to write a number in scientific notation, convert back to standard form, decide the sign of the exponent, and multiply or divide numbers written this way.