e^408 = (e^102)^4.

e^102 = 1.9862648 X 10^44.

Therefore e^408 = (1.9862648 X 10^44)^4 = (1.9862648)^4 X 10^176.

(1.9862648)^4 = 15.56498 = 1.556 X 10.

Therefore e^408 = 1.556 X 10 X 10^176 = 1.56 X 10^177 (to two significant figures).

p.s. Obviously I have a slight disagreement with the text answer.