Lesson 2

Explore

In the Discover section you learned that sometimes you can calculate the root of a negative number depending on what the index of the radical is. When you tried to calculate the square root of −2000, you got an error message. Your calculator may have even displayed a message telling you that the result is a non-real answer. When you work with radical expressions that have variables, it is important to specify restrictions on the variable. All you need to do is write a short mathematical statement that identifies the values of the variable that allow the expression to be simplified or defined.

For example, in the expression , if x is negative, then the radicand becomes negative. Since you can’t find the square root of a negative number, x cannot be negative. On the other hand, if y is negative, the radicand is still positive because any number raised to the fourth power will always be positive. (You can try it on your calculator.) So, the restrictions on the variable are that x cannot be negative and y can be any real number. To state the restrictions on the variable in the expression using mathematical notation, you can simply write x ≥ 0, x ∈ , y ∈ .

If the index of the radical is even, then the radicand cannot be a negative number. For example, all of the following are undefined:

 

 

 

If the index of the radical is odd, then the radicand can be a negative number. For example, all of the following can be simplified: