Regular expression for words containing single 1 and even number of 0s
If a word contains exactly one 1 and an even number of zeroes, then we can decompose it in one of two different ways:


An even number of 0s, followed by 1, followed by an even number of 0s.
An odd number of 0s, followed by 1, followed by an odd number of 0s.


You take it from here.