Dividing powers uses the same counting as multiplying powers, plus one fact you already know: any number other than 0 divided by itself is 1.
Why it works
Write the factors out: $$\frac{x^{7}}{x^{3}} = \frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}$$ Each $x$ on the bottom pairs with an $x$ on top, and each pair is $\frac{x}{x} = 1$. Three pairs cancel, which leaves $7 - 3 = 4$ factors on top: $x^{4}$.
Key idea
To divide powers with the same base, keep the base and subtract the exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n} \quad (a \ne 0)$$
Worked examples
Worked example: Subtract, don’t divide
Write $\dfrac{6^{9}}{6^{4}}$ as one power.
- Four 6s on the bottom cancel four of the nine on top.
- That leaves $9 - 4 = 5$ factors of 6.
Answer: $6^{5}$
Worked example: Numbers and letters
Simplify $\dfrac{15m^{8}}{3m^{2}}$. (Assume $m \ne 0$.)
- The numbers divide like any fraction: $15 \div 3 = 5$.
- The powers of $m$ subtract exponents: $m^{8-2} = m^{6}$.
Answer: $5m^{6}$
Worked example: Small numbers first
Find the value of $\dfrac{2^{10}}{2^{7}}$.
- Subtract the exponents first: $2^{10-7} = 2^{3}$.
- $2^{3} = 8$. Simplifying first keeps the numbers small; you never need to work out $2^{10} = 1{,}024$.
Answer: $8$
Common mistake
Dividing the exponents, as in $\dfrac{x^{6}}{x^{2}} = x^{3}$. Cancelling shows what really happens: 2 pairs disappear and 4 factors are left, so the answer is $x^{4}$. Check with $x = 2$: $\frac{64}{4} = 16 = 2^{4}$, not $2^{3} = 8$.