(sqrt(2)+sqrt(3))^2=9.898979...
(sqrt(2)+sqrt(3))^4=97.989795...(8)
(sqrt(2)+sqrt(3))^8=9601.999896...
(sqrt(2)+sqrt(3))^16=92198401.999999989..
(sqrt(2)+sqrt(3))^32<...4
(sqrt(2)+sqrt(3))^64<...6
so (sqrt(2)+sqrt(3))^100<(sqrt(2)+sqrt(3))^64*(sqrt(2)+sqrt(3))^32*(sqrt(2)+sqrt(3))^4=………………8*4*6=2
so the unit digit is 1
(sqrt(2)+sqrt(3))^4=97.989795...(8)
(sqrt(2)+sqrt(3))^8=9601.999896...
(sqrt(2)+sqrt(3))^16=92198401.999999989..
(sqrt(2)+sqrt(3))^32<...4
(sqrt(2)+sqrt(3))^64<...6
so (sqrt(2)+sqrt(3))^100<(sqrt(2)+sqrt(3))^64*(sqrt(2)+sqrt(3))^32*(sqrt(2)+sqrt(3))^4=………………8*4*6=2
so the unit digit is 1