Exponent Rules

Product adds exponents, quotient subtracts, power of a power multiplies.

The idea

An exponent counts repeated multiplication: $a^{n}$ is the product of $n$ copies of the base $a$, so $a^{3} = a \cdot a \cdot a$. The exponent rules say what happens when such powers are multiplied, divided, or raised to a further power.

Proposition (Exponent rules).

For any nonzero base $a$ and positive integers $m$ and $n$ (with $m > n$ in the quotient), $a^{m} \cdot a^{n} = a^{m+n}, \qquad \frac{a^{m}}{a^{n}} = a^{m-n}, \qquad (a^{m})^{n} = a^{mn}.$

Each rule is a count of copies. In the product, writing $m$ copies of $a$ beside $n$ more gives one product of $m + n$ copies. In the quotient, each copy below cancels one above, leaving $m - n$ copies. In the power of a power, there are $n$ copies of $a^{m}$, each contributing $m$ factors, $mn$ in all.

The product and quotient rules require the same base; $2^{3} \cdot 5^{4}$ combines into no single power. The rules rewrite exponents rather than evaluate anything, and a single power such as $a^{m+n}$ is the finished form.

Ways to work on it

• Walkthrough. All three rules with concrete numbers.
• Practice. Three quick reps — one per rule.
• Hardest. Combine the three rules in one expression.

Not sure where to start? Take the ten-question placement test.