Bean roots will have no nodules if there are no compatible Rhizobium bacteria in the soil. The nodule is a symbiotic relationship between the plant and the bacteria. Nitrogen fixing root nodules are pinkish in color. Green root nodules indicate actively reproducing bacteria that are not fixing nitrogen. The number of nodules depends on the amount of innoculant (Rhizobium) available in the soil. Look at the photos and make an estimate like x nodules per y linear inches of root. Hope this helps.
The edible part of the radish root functions as a food storage organ. The string parts of the radish root function as normal roots absorbing water and dissolved nutrients. Bean roots have no modifications for food storage. Radish roots don't have nodules. The radish root is a "tap root". The bean root is a "fiberous root"
Beans don't grown faster than radishes because of the root nodules. Bean seeds are large. Radish seeds are small. The energy stored in the bean cotyledons helps the seedling get a fast start. The radish gets a slower start from less stored energy.