# 1.4 Roots and fractional indices

To see what fractional indices mean in practice a good place to start is with this example:

Using Rule 3 and 1 this can be written as follows:

This shows that 2^{1/2} is a way to represent the square root of 2. Since .

Extending this *a*^{1/3} is the cube root of *a* and can be written as . *a*^{1/4} is the fourth root of *a*, written as .

So it follows that:

*a*^{1/n} is the *nth* root of *a* and can be written as .

If you use Rule 3 to write *a*^{p/q} as , you should now be able to see that *a*^{p/q} is the *qth* root of *a*^{p} which is written as .

So,

Rules 1, 2, and 3 all work with fractional indices.

Before moving on to another topic, you’ll learn what the power of zero means.