What is the math formula for ?

Here are two numbers:

a = 37

b = 63

Someone’s asking you for (a+b)^5. What do you do?

Lots of people have suggested the binomial formula as an answer to this question. This means

\displaystyle (37+63)^5 = 37^5+5\times 37^4\times 63+10\times 37^3\times 63^2+10\times 37^2\times 63^3+5\times 37 \times 63^4+63^5

which may just be the most horrible way of getting to an answer. Instead, why don’t we just freaking add 37 and 63? That’s actually what (a+b)^5 says we should do first. We get 100. What’s 100^5? Ah, that’s 10^{10}. Ten billion. Done.

If you try to use the binomial formula here, you’ll find yourself doing 37^5=69343957 and 63^5=992463543 and 5 \times 37 \times 63^4=2914297785 and stuff like this, and then adding those things up (yes, one of these is actually wrong. Which one? Did you notice? Would you have noticed?)

That’s terrible. The formula you’ve used was harmful instead of helpful.

So: what is “the math formula” for (a+b)^5?

There isn’t one. I mean sure, there are many, but not one. Not “the” one. (a+b)^5 is already a perfectly fine math formula. Sometimes it is exactly what you need. Sometimes you will definitely need the binomial formula

\displaystyle (a+b)^5 = \sum_{k=0}^5 \binom{5}{k}a^kb^{5-k}

And other times you’ll find yourself using

\displaystyle (a+b)^5 = \exp(5\ln(a+b))

or

\displaystyle (a+b)^5 = a^5\left(1+\frac{b}{a}\right)^5

or

\displaystyle (a+b)^5 = a^5 + b^5 + 5ab(a^3+b^3+2ab(a+b))

or any one of a hundred other versions, variations, expansions and forms.

Things don’t have one single correct formula. Things have shapes and forms, some more useful than others. Sometimes “simplifying” means making things seemingly more complex. The important thing isn’t having formulas, it’s learning how and why you can sculpt expressions to do your bidding.

Formulas don’t do it for you.