Askater holds her arms outstretched as she spins at 130 rpm . what is the speed of her hands if they are 130 cm apart?

This is about tangential vs angular velocity but it is not part of angular momentum which will be explained later on.
Her angular velocity w = 130 rpm = 2pi 130/60 radian/sec. The radius of rotation r = R/2 = 1.3/2 = .65 m from nose to hand. 
Thus the tangential velocity of each hand is v = wr = 2pi*3/2*.65= 6.1261 mps.
Angular momentum L = Iw; where:I = kmr^2 is the inertia and w is the angular velocity. k is a continuous contingent on the shape and density supply of the rotation mass m. So you can see, the problem has zero to do with angular momentum.