The Rule in Letters

The quotient rule written with letters is $\frac{a^{m}}{a^{n}} = a^{m-n}$. It holds for any base $a$ and any exponents $m$ and $n$. The exponent on top minus the exponent on the bottom gives the number of factors that are left.

Look at the right side carefully. The whole difference $m-n$ is the exponent, so when you write it on one line it needs parentheses: a^(m-n). Without them, $a^{m}-n$ means the power $a^{m}$ with $n$ subtracted from it, which is a different expression.

Together with the product rule, $a^{m}a^{n} = a^{m+n}$, you now know the two laws for multiplying and dividing powers with the same base. The next chapter adds a third law, for a power raised to a power.