How to calculate $(a + b)^n$?

Question

I'm trying to review and improve my math using Khan Academy. I'm now beginning to play with derivatives. Calculations like $(x + h)^n$ tend to come up often. I found out empirically that $(a + b)^2 = (a^2 + b^2 + 2ab)$ and a vague memory from school surfaced to confirm this intuition. I'm wondering if there is a method to calculate $(a + b)^n$ for any value of n. If you can also tell me what topic I've missed and should go back to, that would allow me to do do some exercises before carrying on with derivatives.

Answer 1

The formula you are looking for is the binomial theorem.

$$\begin{align}(a+b)^n &= a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n-1}ab^{n-1} + b^n \\ &= \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r} \\&= \sum_{r=0}^{n}\binom{n}{r}a^{r}b^{n-r}\end{align}$$