One thing I've had in the back of my mind for a long time is patterns in end digits for numbers that are powers. I noticed that square numbers never seem to end in 2, 3, 7, or 8 while cubes can end with any number... I moved up to fourth- and fifth-powers and those even-/odd-power patterns seemed to hold.
Do those patterns (for lack of a better word) apply to all powers that are either even or odd, and is there a specific reason why/why not?