a cyclist does work at a rate of 500w while riding. Against what retarding force [friction] is she pushing, if she is traveling at 8 m/s?

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Since this problem is talking about a rate of doing work it is really a power problem.  Power = work / time or P = Fd / t.  F is the force in Newtons,  d is the distance in meters and t is the time in seconds.  Newton meters are Joules and Joules per second are Watts. These are the units that are compatible with the power units given (Watts).  Notice that d / t can be grouped together in the above formula instead of (Fd) / t.  Looked at this way it becomes Force times the speed = the power!  Thus your problem is solved: 500w = F(8 m/s) = 500 J /s = 500 Nm / s (500 Nm / s ) / (8 m/s) = 62.5 N