Exponent Rules: Multiplying, Dividing and Powers of Powers

The three rules

$$a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$$

Every one of them is counting factors. None needs memorising once you have seen why.

Why multiplying adds

Write both powers out.

$$2^3 \times 2^4 = \underbrace{(2 \times 2 \times 2)}_{3} \times \underbrace{(2 \times 2 \times 2 \times 2)}_{4}$$

Removing the brackets leaves seven $2$s in a row:

$$= 2^7$$

The exponents add because you are putting the factors side by side and counting them. Three plus four is seven.

Why dividing subtracts

Division cancels factors in pairs.

$$\frac{2^5}{2^3} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2}$$

Three $2$s on the bottom cancel three on the top, leaving two:

$$= 2^2$$

Five take away three is two. Subtracting counts what survives the cancelling — which is why it is the mirror image of the product rule.

Why a power of a power multiplies

$(2^3)^4$ means four copies of $2^3$:

$$2^3 \times 2^3 \times 2^3 \times 2^3$$

Each copy contributes three factors, and there are four copies:

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

That is why this rule multiplies while the first one adds. Four groups of three, rather than three and four.

$$(2^3)^4 = 2^{12} \qquad\text{but}\qquad 2^3 \times 2^4 = 2^7$$

The bases must match

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

You cannot count $2$s and $3$s together — they are different things. Work each one out separately instead: $8 \times 81 = 648$.

This is the same reason unlike fractions cannot be added until the parts are the same size. Counting needs identical things to count.

A product inside a power

The exponent reaches everything inside the brackets:

$$(2x)^3 = 2^3 x^3 = 8x^3$$

Writing $2x^3$ leaves the $2$ uncubed, which is eight times too small. The same applies to fractions:

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Summary

Rule	Form	Because
product	$a^m a^n = a^{m+n}$	factors 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

Worked examples

Common mistakes

Practice problems