To add or subtract radical expressions, simplify first and then add like terms if there are any. The radical parts of the terms must be exactly the same before we can add them. (Assume all variables are positive.)
Perform the operations.
Be sure to follow the correct order of operations.
Simplify.
Unsimplified rational expressions may look as if they have no like terms, but first try simplifying and then check for like terms.
Simplifying is a challenge for many students. In particular, the numerical part is the source of confusion. It is much easier to deal with the prime factors of the number in a radical than it is with the number itself. Create a factor tree and determine the prime factorization of all numbers before simplifying radical expressions.
Simplify:
Begin by determining the prime factorizations of 108 and 864.
Substitute the prime factorization in and then simplify.
The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value.
To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. If not, use the absolute values as in the following problems.
Simplify. (Assume variables could be negative.)
To avoid many of the technicalities when working with nth roots we will assume, from this point on, that all the variables are positive.
Simplify. (Assume all variables represent positive numbers.)
Students often try to simplify the previous problem. Be sure to understand what makes the following problems all different.
The property:
says that we can simplify when the operation is multiplication. There is no corresponding property for addition or subtraction.
