# Roots and Radicals

## Add and Subtract Square Roots

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify  in this way:

So if we have to add , we must not combine them into one radical.
Trying to add square roots with different radicands is like trying to add unlike terms.

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.
We add and subtract like square roots in the same way we add and subtract like terms. We know that  is . Similarly we add  and the result is 

### Add and Subtract Like Square Roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.
When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

### Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.
Just like we use the Associative Property of Multiplication to simplify  and get , we can simplify  and get . We will use the Associative Property to do this in the next example.
In the next example, we will remove constant and variable factors from the square roots.

### Key Concepts

1. To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
2. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

### Practice Makes Perfect

Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
Mixed Practice

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?