The Rule in Letters

The power of a power rule written with letters is $\left(a^{m}\right)^{n} = a^{mn}$. It holds for any base $a$ and any exponents $m$ and $n$. Each copy of $a^{m}$ holds $m$ factors of $a$, and there are $n$ copies, so there are $mn$ factors of $a$ in all.

For example, with $a = x$, $m = 3$ and $n = 2$, the rule gives $\left(x^{3}\right)^{2} = x^{6}$.

You now have three laws for exponents with the same base. The product rule, $a^{m}a^{n} = a^{m+n}$, is for two powers multiplied side by side, and the exponents add. The quotient rule, $\frac{a^{m}}{a^{n}} = a^{m-n}$, is for one power divided by another, and the exponents subtract. The power of a power rule, $\left(a^{m}\right)^{n} = a^{mn}$, is for a power inside a power, and the exponents multiply. All three come from counting factors of the base.