A Pattern for Fractional Indices

We will now look at the meaning of using patterns. We will need a new pattern. Here is one which seems to lead naturally to the indices of half and a quarter. What do you think the pattern is? Remember to describe the pattern in terms of mathematical operations.

What is the pattern in this table?

Fourth Roots

If we multiply by twice in order to multiply by three once, then perhaps we need to miltiply by four times in order to multiply by three once. This would look like this:

But this is the same idea as a fourth root, which suggests that . But the pattern in the table suggests that . Are these things the same?

Is ?

Roots As Indices

We can now generalize our ideas. It seems that fractional indices lead to the same ideas that we have been calling roots. Here is the formal statement of this idea.

If is any number, , and is any integer, , then

Note that since it is not always possible to take a root of a negative number, and since roots are only defined for positive integers.