[math-fun] Multiplying repeating decimals

I had assumed that if you multiply an eventually repeating decimal of period m and an eventually repeating decimal of period n, you get an eventually repeating decimal whose period is bounded by some polynomial function of m and n. But today I learned from Henry Cohn that that's not true: the period length can be exponentially large in m and n. More specifically, 1/(10^n-1) repeats with period n, but its square repeats with period 10^n-1 or thereabouts. I'm sure someone has written about the repeat-length for 1/(10^n-1)^2; can anyone provide a reference? Jim Propp