To have a zero at the end, a number must be multiplied by 10. Another way to think of this is that the number must be multiplied by the prime factors 5 and 2.

135 can be factored into 5 and 27. There are no other factors of 5 or 2.
42 can be factored into 2 and 21. There are no other factors of 5 or 2.

So 135^19 will have 19 5s, and 42^17 will have 17 2s. So the answer will end in 17 zeros because there are more 2s than 5s.