Lesson 2
1. Lesson 2
1.7. Explore 3
Module 5: Radicals
Another way to convert to a mixed radical is to completely factor the radicand into prime factors.
Self-Check 2
Change the following entire radicals into mixed radicals, and state any restrictions on the variables.
The radical sign represents the principal root of a number. Even though (−4)² = 16, mathematicians have agreed that the radical symbol results in a positive number, so 
. If the secondary root is wanted, it must be specified as 
. If both principal and secondary roots are wanted, they should be specified as 
.
The radical 
is only defined as a real number if x is greater than or equal to zero. In other words, it cannot be a negative number. However, the value of 
is defined as a real number for all values of x because x2 will be positive whether x is positive or negative.
To convert a mixed radical to an entire radical, you apply the same process whether you have variables in the expression or not.
Try This 3
Change the mixed radical 
into an entire radical.
- Write each term of the coefficient under the radical sign with the index as its exponent. Answer
- Arrange all numerical factors to have an exponent of 1 and find the resulting product. Answer
- Check your work. Substitute a value for the variable in the mixed radical and the entire radical, and use your calculator to verify that the results are the same. Answer
The same procedure can also be used for cube roots. Change the mixed radical 
into an entire radical.