There is a connection between exponents and roots that will be explored in later lessons. To understand the connection, it is helpful to know some exponent rules and to understand roots in terms of powers that "undo" them.

We can understand as a lot of s multiplied together. In particular, there are of them multiplied together and then another of them multiplied together. This means that there are a total of  of the s multiplied together, which can be written as .

Similarly, we can understand as groups of , all multiplied together. That means there are a total of of the s multiplied together, which can be written as .

When dividing expressions involving exponents, like , let’s consider two situations. 

First, let’s look at the situation in which there are at least as many s in the numerator as in the denominator (). We can separate out of the s from the numerator and still have of them left. This can be written as . By rewriting the fraction, we can get which is the same as because the first fraction has a value of 1.

Second, let’s look at the situation in which there are more s in the denominator than in the numerator (). Again, we can separate the fraction into , or . Then if we recall the meaning of a negative exponent, we can see that , which is the same as again.

Here are all of the exponent rules we looked at in this lesson:

It is also helpful to recall the meaning of . We can say that is a solution to the equation (when ).