Not sure if cycle is the best word here.. Let k and b (both > 1) be such that b does not divide k-1 (with the exception that b may equal k-1). If n is the smallest power so that k^n is equivalent to 1 mod b (so n is the length of the digit cycle of 1/b in base k), then bn is the smallest power so that k^(bn) is equivalent to 1 mod b^2 (so bn is the length of the digit cycle of 1/(b^2) in base k). In base 10, 1/81 has length 9. 1/9 has length 1. I've checked many other examples.