Suppose a star the size of our sun, but of mass 9.0 times as great, were rotating at a speed of 1.0 revolution every 17 days. if it were to undergo gravitational collapse to a neutron star of radius 15 km, losing 3/4 of its mass in the process, what would its rotation speed be? assume the star is a uniform sphere at all times

Use the conservation of angular momentum; angular momentum at the beginning = angular momentum at the end 
Conservation of angular momentum: 
I1 w1 = I2 w2 
Where I is the moment of inertia. For a sphere, I=2/5 m R^2. Substituting into the equation above we get 
w2 = I1 w1 / I2 = w1 m1 R1^2 / (m2 R2^2) 
w2 = w1 4 * (R1/R2)^2
= 4*(1)*(7E5/7.5)^2
= 3.48E10 revs/(17days)
= 2.04705882 x 10^9 revs/sec