Imagine that you have a long pipe to begin with. For each stage you are losing 15% of the impurities which means you have (1-0.15) remaining to go onto the next stage. At the next stage you no longer have all the impurities but the (1-0.15) from the previous stage, since another 15% or (1-0.15) is lost then (1-0.15)*(1-0.15) = (1-0.15)^2 remains.

This continues until you get to thexth stage where you will have (1-0.15)^(x-1) * (1-0.15) = (1-0.15)^x. As 1-0.15 = 0.85 you arrive at the answer you mentioned of 0.85^x.

You're not subtracting the 0.15 from the first stage if that makes sense, imagine the mixture has travelled 0 feet and thus there would be (1-0.15)^0 = 1, ie: no impurities have gone which is what we'd expect

Interestingly, due to the nature of exponential equations all of the impurities cannot be filtered out in this manner.