I've been playing with a new math observation. It started when I noticed the following sequence for the nth roots of 2: (1 + 7⁄17)2 ≈ 2;
(1 + 7⁄27)3 ≈ 2; (1 + 7⁄37)4 ≈ 2; (1 + 7⁄47)5 ≈ 2. This can be generalized to (1 + 7⁄(10x-3))x = ((10x + 4)⁄(10x-3))x ≈ 2.
So, is this an isolated fluke, or are there expressions like this for all roots? It turns out there are multiple expressions (1 + a⁄(bx-c))x which converge to any N for large x. For 2, the combinations (2, 3, 1), (5, 7, 2), (7, 10, 3) and (9, 13, 4) give increasingly closer convergence to 2. The ratio a⁄b seems to be about .694, but I haven't yet mastered the limit expressions to get a symbolic expression for the number. I've found through trial and error combinations for roots of 3, 4, 5, and 10 - however, the convergence seems slower for larger N, meaning that the approximations are very poor for the lower roots (those you're more likely to use).
So, I haven't yet determined if this discovery is significant or trivial, but if you should happen to be at the sort of party where people are impressed by mental math (I haven't yet found one), you can rattle off that the 7th root of 2 is roughly 1 + 9⁄87. Or, more likely, you'll just have a handy small-number ratio for a root if you're doing math by hand.