Key Lesson Marker


When the index of a radical is even, the radicand must be greater than or equal to zero:

However, when the index of a radical is odd, the radicand can be positive, negative, or zero:

Example 7

Evaluate the following expressions, if possible. If the expression cannot be evaluated, explain why.




  1. The expressions cannot be evaluated. The square root of a negative number cannot be evaluated with the Real Number system. The index of 2 indicates a number, that when multiplied by itself gives the radicand, is needed. However, a number multiplied by itself will always be positive. Since the radicand in this case is negative, the expression cannot be evaluated.





  2. Since the negative sign is on the outside of the brackets, the radical can be evaluated (since it is the square root of a positive number) and then multiplied by –1.





  3. The cube root of a positive or negative number can be taken. The index of 3 indicates a number, that when multiplied by itself 3 times gives the radicand, is needed. A number multiplied by itself 3 times can be positive or negative and thus the given expression can be evaluated as shown.

With the tools covered in this unit, along with what you have previously learned in other math courses, it is expected that you will work to simplify and evaluate expressions involving radicals and exponents without the use of a calculator. When verification of a solution or an approximate (rather than exact) value is required, a calculator is a valuable tool to confirm what you have already calculated by hand.