A curious square root (Part 1)

Here’s a square root that looks like a total mess:

Just look at this monstrosity, which has a triply-embedded square root! But then look what happens when I plug into a calculator:

Hmmm. How is that possible?!?!

I’ll give the answer after the thought bubble, if you’d like to think about it before seeing the answer.

Let’s start from the premise that and work backwards. This isn’t the best of logic — since we’re assuming the thing that we’re trying to prove in the first place — but it’s a helpful exercise to see exactly how this happened.

This last line is correct, using the formula . So, running the logic from bottom to top (and keeping in mind that all of the terms are positive), we obtain the top equation.

This suggests a general method for constructing such hairy square roots. To begin, start with any similar expression, such as

Then we create a nested square root:

Then I get rid of the square root on the left hand side:

Then I add or subtract something so that the left-hand side becomes a perfect square.

Finally, I take the square root of both sides:

Then I show the right-hand side to my students, ask them to plug into their calculators, and ask them to figure out what happened.