Expanding
October 15, 2024 at 10:41 AM by Dr. Drang
Regular readers^{1} know that I like to watch mathish YouTube videos from Numberphile, Standup Maths, and Mind Your Decisions. Unfortunately, this leads YouTube’s machine learning system to believe that I want to watch all math videos, no matter the level or quality. So it keeps pushing dull videos at me in which an overhead camera records the hands and pen of the presenter writing out line after line of elementary math. There are apparently an infinite supply of these things.
Occasionally, one of these videos has something in the thumbnail frame that catches my eye, and I click on it despite myself. Here’s a video I saw last night that pretends there’s a connection between a binomial expansion and Stephen Hawking. What it really does is solve what it says is a problem from a high school entrance exam. No Hawking at all.
The goal is to simplify this expression:
$$(\sqrt{2}1{)}^{10}$$It could be argued that this is already in a pretty simple form—it’s certainly compact—but in this case, “simplify” means to get rid of the exponent.
If you can bear the tedium of the video, you’ll see that its solution goes through some algebra and various levels of substitution. It takes a very long time, and not just because the presenter insists on writing out every little step along the way.^{2} The entire approach to solving the problem is inefficient.
Because this is an exam problem, getting the solution quickly is important. To my mind, the fastest solution is to expand the binomial directly. I don’t have the binomial coefficients memorized up to the 10th degree, but Pascal’s Triangle can be built in a jiffy:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
You know how to put this together, right? Each new row starts and ends with a 1, and the other terms come from the sums of the terms in the previous row diagonally to the left and right, e.g.,
$$10=1+9$$ $$45=9+36$$ $$120=36+84$$ $$210=84+126$$and so on. Because of the symmetry, you only have to calculate about half the terms.
So the expansion of $(\sqrt{2}1{)}^{10}$ is
$${2}^{5}10\cdot {2}^{4}\sqrt{2}+45\cdot {2}^{4}120\cdot {2}^{3}\sqrt{2}+210\cdot {2}^{3}$$ $$\phantom{\rule{thinmathspace}{0ex}}252\cdot {2}^{2}\sqrt{2}+210\cdot {2}^{2}120\cdot 2\sqrt{3}+45\cdot 210\sqrt{2}+1$$where I’ve used identities like
$${\left(\sqrt{2}\right)}^{10}={2}^{5}$$and
$${\left(\sqrt{2}\right)}^{9}={2}^{4}\sqrt{2}$$to simplify each term in the expansion.
Now we can do the various multiplications and collect the terms with and without the $\sqrt{2}$ terms to get
$$(32+720+1680+840+90+1)$$ $$(160+960+1008+240+10)\sqrt{2}$$which gives us the correct answer:
$$33632378\sqrt{2}$$While there may be some cleverness to the YouTuber’s method of substitution and resubstitution, this is certainly more straightforward and less likely to generate algebra mistakes—both of which are important when you’re under pressure during a test. The main thing to remember is that the odd powers of $\sqrt{2}$ have negative signs and the even powers don’t.
I suppose it’s unfair for me to insult someone who’s trying to help students, but his slow solution isn’t going to be very helpful. And he shouldn’t pretend that his video has anything to do with Stephen Hawking.

Can you be a regular reader of a blog as irregularly published as this one? Discuss. ↩

Yes, I realize that this is an instructional video meant for young people and there’s value in a methodical pace. But no one who can follow the math in this video needs to see every single addition and multiplication. ↩