Say you want to find the units digit of .
So you are working with mod 10.
But how do we get ?
What sequence of steps did we use to get to mod 10?
Divide top and bottom by to get . Then use the binomial expansion
The terms before the highlighted one all give multiples of 10, so will not affect the units digit. The terms following the highlighted one are all very small and rapidly decreasing fractions, and the first of them is positive, so the overall sum of them will be a small positive fraction. Thus the highlighted term is the one that determines the units digit.