In this explainer, we will learn how to add and subtract radical expressions.
Before we start looking at how to add and subtract radicals, let us first clearly define what we mean when we talk about radicals and define some associated mathematical language.
Firstly, the word radical describes the “root sign,” the number contained within the root sign is called the radicand, and the little number on the outside of the radical is called the index, which tells us if we are dealing with a square root, a cube root, or otherwise. These terms have been labeled in the following picture:
We can add or subtract two radical expressions if and only if they both have the same index and the same radicand. We need to be careful not to confuse the index of a radical with the coefficient. For example, means the cube root of 8 which is 2, whereas means . In the first case, the 3 is the index and in the second case it is the coefficient. So, let us look at a couple of examples of cases where we can and cannot simplify the expressions. First, consider
Both of the radicals are square roots and they both contain the same radicand. We have and we are adding so we have a total of . Now consider the expression
Here, both radicals have the same radicand but they have different indices and, thus, cannot be simplified. Again, be careful here as this example is often mistakenly simplified to . Finally, consider the expression
These two radicals have the same index but different radicands so, consequently, they cannot be simplified. This, however, is not the end of the story as there are some expressions where it may look like they cannot be simplified but the radicals themselves can be simplified to minimize their radicand, and following this, the two expressions can be combined. We will not go into too much detail of this here, but an example of this would be the expression
The two radicals have the same index, but the radicands are not the same. We can, however, rewrite :
We can then rewrite the expression as which simplifies to .
Therefore, when asked to simplify the sum or difference of radicals always do the following checks.
Checklist for Adding and Subtracting Radicals
- Check that the radicals have the same index.
- Check that the radicals have the same radicand.
- If the radicals have the same index but different radicands, check to see if either or both of the radicals can be reduced to the point where they have the same radicand.
Let us now look at some formal examples of questions from this topic.
Example 1: Finding the Sum or Difference of Radical Expressions
Firstly, we need to check that all of the radicals have the same index and that the radicands are the same. For this question, this is true. The coefficients of all the radicals are one, so he have that
Now, let us look at a few more slightly more difficult examples where the expression contains terms that are radicals and terms that are not. For example, if we consider the expression a very common error is to simplify this to whereas, in reality, the expression cannot be simplified further.
Example 2: Finding the Sum or Difference of Radical Expressions
A sensible first step is to rewrite the expression to separate the radical and nonradical terms:
Then, verify that the radical terms have the same index and radicand. Once this is confirmed, we can combine the like terms to get
Example 3: Finding the Sum or Difference of Radical Expressions
A sensible first step is to rewrite the expression using properties of addition:
Then, verify that the radical terms have the same index and radicand. Once this is confirmed, we can combine the like terms to get 6. Note here that , which is why our solution is just 6.
To finish, let us look at an example where it is not immediately obvious that the expression can be simplified.
Example 4: Simplifying Expressions Containing Multiple Radicals
Here, the radicals all have the same index but the radicands are different, so it is not immediately clear that they can be simplified. However, two of the three radical terms are reducible, namely, and . We can reduce these by rewriting the radicands as products of a square number. We can simplify as follows to get and we can simplify to get
We can then rewrite our original expression to get which simplifies to .