If x= 1/(2^2*3^2*4^2*5^2) is expressed as a decimal, how many distinct

You don't really need to do long division here if you don't want to, as long as you know that 1/9 = 0.11111....., and 2/9 = 0.22222...., and so on.

If we multiply a number by 10, or by 100, or by any other power of 10, we're just moving the decimal point over. We won't change the number of distinct nonzero digits we have in our number. So we can freely multiply the fraction by any power of 10 without changing the answer. We can start by multiplying the fraction by 10^2 = 2^2 * 5^2, to get:

1/3^2*4^2

which is already a bit simpler. But we can keep going - we can multiply by 100^2 = (4*25)^2 = 4^2 * 25^2 to get the 4s out of the denominator as well. So this fraction

25^2/3^2 = 625/9

has the same number of distinct digits as the fraction in the question. Since 630/9 = 70, then 621/9 = 69, so 625/9 = 69 + 4/9 = 69.44444..... which has three distinct digits.

But I think most people would just do long division, either with 1/144, or with 1/14,400, which will also work, and probably is just as fast. I hate doing long division so I always look for another way!