Further Extending the Definition

In the work on positive and negative indices we manged to find a meaning for powers with an integer index, whether this index was positive, zero or negative. We will now continue the method of looking at definitions, patterns and properties to find a meaning for fractional indices.

We will start with something very simple. We will look for a meaning for .

First Definition

Our first definition says that is an expression with half of the factor three. How can you have half a factor? Well so far we have managed to find a negative number of factors so why not half. Let's give it a go.

If it were possible to have half of something then you would expect that having two of them (that is two lots of half of something) would be the same as having one of them. So if we have two lots of this should be the same as having a single factor three. Now how should we combine factors. Since factors involve multiplication perhaps we should combine these two half factors by multiplying them together. This would give us the equation.

This looks good because straight away you can see that it agrees with our rule for adding indices with integers.

Second Definition

Our second definition says to start with 1 and multiply by 3 half a time. How can you do something half a time? Well perhaps it means that in order to multiply by three once we have to multiply by three half a time twice. Confusing? I guess it would look like this.

This seems to lead to the same result as definition 1.

Isn't this a Square Root?

A square root of three is defined to be a number which when multiplied by itself gives three. In equation form this looks like this.

So it lloks as if an index of one half has something to do with square roots. Can it be that: