TITLE: How to simplify $a^n - b^n$?
QUESTION [10 upvotes]: How to simplify $a^n - b^n$?
If it would be $(a+b)^n$, then I could use the binomial theorem, but it's a bit different, and I have no idea how to solve it.
Thanks in advance.

REPLY [4 votes]: $$
a^n -b^n=a^n\Big[1-\Big(\displaystyle\frac{b}{a}\Big)^n\Big]
$$
If $a=b$,  then $a^n -b^n$ is equal to zero and there is no need for a formula. Suppose $a\neq b$ and $\displaystyle\frac{a}{b}\neq 1$. Then we can use the formula for the computation of a finite geometric series which is:
$$
\sum_{i=0}^{n-1}r^i=\frac{1-r^n}{1-r},\quad r\neq 1
$$
Let $r=\displaystyle\frac{b}{a}$ in the above formula. This gives us:
$$
\begin{align}
a^n-b^n&=a^n\Big[\Big(1-\frac{b}{a}\Big)\sum_{i=0}^{n-1}\Big(\frac{b}{a}\Big)^i\Big]=a\cdot a^{n-1}\Big[\Big(1-\frac{b}{a}\Big)\sum_{i=0}^{n-1}\Big(\frac{b}{a}\Big)^i\Big]\\
&=(a-b)a^{n-1}\Big[\sum_{i=0}^{n-1}\Big(\frac{b}{a}\Big)^i\Big]\\
&=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+ ... +ab^{n-2}+b^{n-1})
\end{align}
$$