Maybe this isn't the slickest response, but sometimes you just have to go with what you know, and get to the answer fast. For questions like this - I usually like to see if I can quickly establish a pattern. Also - I hate fractions. So a lot of the times, I convert to a decimal.
1/5 = .2
(1/5)^2 = .2 x.2 =.04
(1/5)^3 = .04 x.2 =.008
(1/5)^4 = .008 x.2 =.0016
(1/5)^5 = .0016 x.2 =.00032
so I quickly see I am counting by 2s and the last digit of the number winds up in the "place" of the power. Looking for patterns IS powerful, especially on this exam.
So, I use my fingers (no kidding! and I have a PhD!) to double the number, up to 10 times. The number is 1024. So, according to the pattern, there is a 4 in the 10th place after the decimal.
The GMAT is less testing how well you do math, it is testing how quickly you can solve problems.
I'm sure there are other ways to solve the problem. The trick is practicing different ways so you can be quick and nimble on test day.
