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.


Fraction identity. 


Third law of indices 


Definition of indices as roots. 
This could have been done slightly differently:


Fraction identity. 


Third law of indices 


Definition of indices as roots. 
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:
