The speed of a wave pulse on a string depends on the tension, f, in the string and the
mass per unit length, µ, of the string. tension has si units of kg • m • s-2 and the mass
per unit length has si units of kg • m-1. what combination of f and µ must the speed of
the wave be proportional to?

Speed of a wave is in m * s-1, so we can figure this out through process of elimination to make the units of Tension and mass/length equal our goal units of m/s.

First, we can see there are no kg in our result, so no matter what we have to have F/μ or μ/F so we get our kg to cancel out.
Now we are left with either: 

/F

or

 

Now if our goal is m/s, the former of F/μ is closer, but off by a square. If we correct that by squaring F/μ we get