Those familiar with algebra definitely knows about the expansion rule when dealing with multiplication of terms involving sums on the inside. For example, the multiplication of (a+b) and (c+d) has the following product:
(a+b)(c+d) = ac + ad + bc + bd
The expansion rule states that you have to distribution out the terms when multiplying them together. Another way to look at it is in the expansion rule, the expansion is the sum of all the combination of each terms in the product.
So in the above example, the terms on the right hand side is the sum of the combinations of a and b from (a+b), and c and d from (c+d); which are ab, ac, bc and bd.
Expansion rule is something that after using it for a while it becomes so natural that we just accept it and don't really think about it.
To become good with mathematics it is important to not just accept rules like that, it also involves examining things and relentless ask why to seek deeper understanding. Let us take a step back and ask: why does the expansion rule work?
There is a nice and simple visual way to explain this. Consider the following rectangle with sides of a+b and c+d:
Finding (a+b)(c+d) is essentially the same as finding the area of the above rectangle. By breaking down each sides into two with lengths of a, b, c and d respectively; it is now apparent why the expansion rule works.
The total area can be found by adding up the areas of each of the smaller rectangles, which are, none other than: ac, ad, bc and bd!
So there you have it, a simple yet elegant explanation for the expansion rule! Of course this is not the only way to explain it, as there are a few others too!