Using the Properties

We have only two jobs left to complete our story of indices. So far all our fractional indices have been the reciprocals of integers. We need to generalize this to any fractions. We also need to mix up fractions with negative indices. This time we will use the properties of indices to tell us what these things might be. This is how we might use the third law of indices.

This could have been done slightly differently:

The only difference between these two is the order of operations. Since multiplication can be done in any order, the two resulting expressions are equivalent. In fact there are six different ways of writing and here they all are: