In $a^n$, $a$ is the base and $n$ is the index. The diagram shows how a fractional index splits into a root and a power, while a negative index tells you to take a reciprocal.

The key laws are below.

$$a^m\cdot a^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n},{\left(a^m\right)}^n=a^{mn}.$$

These work when the base is the same. These are not separate topics: they are all part of the same pattern of powers. A quick memory aid is denominator = root, numerator = power.