What is the angular speed of Earth, in radians per second, as it rotates about its axis?

The key to this problem is recognizing that one revolution is 2pi radians.  The Earth completes one rotation (which is a revolution about an internal axis) every 24 hours.  We start by converting 24h into seconds:24h *3600s/h=86400s

The unit of hour cancels and we are left with seconds.
Then to get angular velocity, we compare the radians covered by the time taken.  So we divide 2pi rad by the time in seconds:
(2pi(rad))/(86400s)=0.0000726(rad)/s

And we have our answer:  7.26x10^-5 rad/s

The key to this problem is knowing that the Earth rotates through 2π radians every 24 hours. Then apply the definitions of angular speed.
http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html