Question

Find the length of both arc AB and arc APB, where

Answer 1

check the picture below.

as you can see, the angle at P is the inscribed angle, and the intercepted arc is AB, thus 2P = arcAB.

and since a circle has a total of 360°, the other arc is just 360 - arcAB.

Answer 2

The length of the arc cannot be found without knowing some linear measure of the circle. None are given.

The measure of arc AB is twice that of angle APB, so is
  2*108° = 216°.
The measure of arc APB is the difference between that and 360°, so is
  360° -216° = 144°.