What's the source of the approximation $\pi\approx\sqrt [4]{9^2+\frac{19^2}{22}}$?

Question

I found this approximation of $\pi$ in my math textbook, and I don't know where it comes from (or even how the person came up with this): $$\pi\approx \sqrt [4]{9^2+\frac{19^2}{22}}$$ Can someone tell me where this is from and your thoughts on how the person came up with this? Thanks.

Answer 1

Writing as continued fraction:

$$ \pi^4 =[97;2,2,4,-16541,\dots] $$

This means stopping expansion after 4 gives a really good rational approximation, namely

$$\pi^4 \approx 97+\frac9{22}$$

Rest is writing it more neatly as $\pi^4\approx 9^2+\cdots$