This is a 30 - 60 - 90 triangle in the diagram. You are told that because of the right angle and another measuring 60 degrees. Since each triangle measures 180 degrees for all its angles, then ∠M is 30 degrees.
In a 30 - 60 - 90 triangle, the pattern of the legs is as such:
short leg: 1 unit
long leg: √3 units
hypotenuse: 2 units.
We can conclude that the hypotenuse is double the short leg, and we use that here.
We are told the hypotenuse is 8 mm, so the short leg is going to be 4 mm. (The opposite of doubling is halving, hence why divide by two, not multiply.
Now this gives us NP, not MN. We want that, but because we know the short leg and the triangle type, we can figure that. Go back to the 1 - √3 - 2 pattern. If the short leg is 4, we multiply it by √3 to get the long leg.
Thus, the long leg is 4√3 mm.