Adding and Subtracting Radicals
Learning Objective(s)
· Add radicals and simplify.
· Subtract radicals and simplify.
Introduction
There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals.
Making sense of a string of radicals may be difficult. One helpful tip is to think of radicals as variables, and treat them the same way. Let’s start there.
Thinking about Radicals as Variables
Radicals can look confusing when presented in a long string, as in _{}. How do you simplify this expression? (It is worth noting that you will not often see radicals presented this way…but it is a helpful way to introduce adding and subtracting radicals!)
Treating radicals the same way that you treat variables is often a helpful place to start. For example, you would have no problem simplifying the expression below.
_{}
Combining like terms, you can quickly find that 3 + 2 = 5 and a + 6a = 7a. The expression can be simplified to 5 + 7a + b.
The same is true of radicals. As long as radicals have the same radicand (expression under the radical sign) and index (root), they can be combined. Below, the two expressions are evaluated side by side.
_{}  _{} 
_{}  _{} 
_{}  _{} 
Here’s another way to think about it. Recall that radicals are just an alternative way of writing fractional exponents. So, for example, _{}, and _{}. If you think of radicals in terms of exponents, then all the regular rules of exponents apply.
Look at the expressions below. On the left, the expression is written in terms of radicals. On the right, the expression is written in terms of exponents.
_{}  _{} 
_{}  _{} 
_{}  _{} 
So what does all this mean? Well, the bottom line is that if you need to combine radicals by adding or subtracting, make sure they have the same radicand and root. And if things get confusing, or if you just want to verify that you are combining them correctly, you can always use what you know about variables and the rules of exponents to help you.
Let’s look at some examples. In this first example, both radicals have the same root and index.
Example  
Problem  Add. _{} 

 _{}  The two radicals are the same, _{}. This means you can combine them as you would combine the terms _{}. 
Answer  _{} 

This next example contains more addends. Notice how you can combine like terms (radicals that have the same root and index) but you cannot combine unlike terms.
Example  
Problem  Add. _{} 

 _{}  Rearrange terms so that like radicals are next to each other. Then add. 
Answer  _{} 

Notice that the expression in the previous example is simplified even though it has two terms: _{} and _{}. It would be a mistake to try to combine them further! (Some people make the mistake that _{}. This is incorrect because_{} and _{} are not like radicals so they cannot be added.)
Example  
Problem  Add. _{} 

 _{}  Rearrange terms so that like radicals are next to each other. Then add. 
Answer  _{} 

Sometimes you may need to add and simplify the radical. If the radicals are different, try simplifying first—you may end up being able to combine the radicals at the end, as shown in these next two examples.
Example  
Problem  Add and simplify. _{} 

 _{}  Simplify each radical by identifying perfect cubes. 
 _{} 

 _{} 

 _{}  Simplify. 
 _{}  Add. 
Answer  _{} 

Example  
Problem  Add and simplify. _{}  
 _{}  Simplify each radical by identifying perfect cubes. 
 _{} 

 _{} 

 _{}  Add like radicals. 
Answer  _{} 

Add. _{}
A) _{} B) _{} C) _{} D) _{}

Subtraction of radicals follows the same set of rules and approaches as addition—the radicands and the indices (plural of index) must be the same for two (or more) radicals to be subtracted. In the three examples that follow, subtraction has been rewritten as addition of the opposite.
Example  
Problem  Subtract. _{} 

 _{}  The radicands and indices are the same, so these two radicals can be combined. 
Answer  _{} 

Example  
Problem  Subtract. _{} 

 _{}
_{}  Two of the radicals have the same index and radicand, so they can be combined. Rewrite the expression so that like radicals are next to each other. 
 _{}  Combine. Although the indices of _{} and _{} are the same, the radicands are not—so they cannot be combined. 
Answer  _{} 

Example  
Problem  Subtract and simplify. _{}  
 _{}  Simplify each radical by identifying and pulling out powers of 4. 
 _{} 

 _{} 

Answer  _{} 

Subtract and simplify. _{}
A) _{} B) _{} C) _{} D) _{}

Summary
Combining radicals is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables: like radicals can be added and subtracted in the same way that like variables can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.