Just like learning to add fractions and decimals, there are rules for adding and subtracting surds.
Fortunately, the process is actually quite simple. If you know what a surd is and how to simplify surds then you’re ready to learn how to add and subtract surds.
How to Add and Subtract Surds
What is a surd?
As a quick reminder, a surd is any root whose value is an irrational number. An irrational number is a number which cannot be written as a fraction. The digits after the decimal point continue forever and never repeat.
Any root whose value is not a whole number is a surd.
| Surd | Not a Surd |
|---|---|
| √2 = 1.414213562… | √4 = 2 |
| √13 = 3.605551275… | √81 = 9 |
| √27 = 5.196152423… | √0 = 0 |
| 3√3 = 1.44224957… | 3√1 = 1 |
| 3√21 = 2.758924176… | 3√27 = 3 |
| 4√20 = 2.114742527… | 4√81 = 3 |
To see how to add and subtract surds, first look at this algebraic expression.
2x + 3x
We know that we can simplify the terms of this addition as they are like terms.
2x + 3x = 5x
If we say that x = √7 then 2x + 3x = 2√7 + 3√7.
We say that 2√7 and 3√7 are like surds as they both have the same number inside the root.
The number inside of the root is called the radicand.
2√7 and 3√7 are called like surds because they both have a radicand of 7.
Just as we can add like terms together, we can also add like surds together.
So in this case, 2√7 + 3√7 = 5√7.
In general, when adding or subtracting surds we can only simplify if they are like surds.
We can simplify 6√5 + 9√5 because 6√5 and 9√5 are like surds; they both have a radicand of 5.
6√5 + 9√5 = 15√5
We cannot simplify 6√5 + 9√4, because they are not like surds; one has a radicand of 5 and one has a radicand of 4.
Adding and Subtracting Surds Examples
Example 1
Question: √8 + 4√8
√8 and 4√8 are like surds as they both have a radicand of 8. We can simplify them.
Note that √8 is the same as 1√8
√8 + 4√8 = 5√8
Answer: 5√8
Example 2
Question: 3√13 + 4√13
3√13 and 4√13 are like surds as they both have a radicand of 13. We can simplify them.
3√13 + 4√13 = 7√13
Answer: 7√13
Example 3
Question: 8√3 – 6√3
8√3 and 6√3 are like surds as they both have a radicand of 3. We can simplify them.
8√3 – 6√3 = 2√3
Answer: 2√3
Example 4
Question: 12√3 – 12√2
12√3 and 12√2 are not like surds as the first has a radicand of 3 but the second has a radicand of 2. We cannot simplify them.
Answer: 12√3 – 12√2 (already simplified)
Example 5
Question: 3√8 – 9√8 + 2√8
This expression contains 3 terms but they are all like surds so we can simplify them. First subtract 9√8 and then add 2√8.
3√8 – 9√8 = -6√8
-6√8 + 2√8 = -4√8
Answer: -4√8
Example 6
Question: 2(4√3) + √3
This example also requires multiplication because 2(4√3) = 2 × (4√3).
Imagine multiplying 2 × (4x).
2 × (4x) = 8x and in the same way 2 × (4√3) = 8√3.
Back to the original question:
2(4√3) + √3 = 8√3 + √3 = 9√3
Answer: 9√3
So far, all of the examples have been fairly simple. Sometimes however, there can be an additional step before adding or subtracting.
Simplifying Surds Before Adding or Subtracting
Consider the example √8 + 3√2.
At first glance it seems like we cannot simplify this expression.
However, let’s try to simplify √8 first.
This means that √8 + 3√2 is the same as 2√2 + 3√2.
After simplifying, we can see that √8 and 3√2 are like surds even though they don’t appear so at first.
As they are like surds, we can now add them.
√8 + 3√2 = 2√2 + 3√2 = 5√2
In order to add or subtract surds, we need to make sure they are both simplified first.
Simplifying First Examples
Example 1
Question: √3 + √12
√3 is already simplified.
√12 = √(4 × 3) = √4 × √3 = 2 × √3 = 2√3
So √3 + √12 = √3 + 2√3 = 3√3
Answer: 3√3
Example 2
Question: √150 – √6
√150 = √(25 × 6) = √25 × √6 = 5 × √6 = 5√6
√6 is already simplified.
So √150 – √6 = 5√6 – √6 = 4√6
Answer: 4√6
Example 3
In this example we have to simplify both terms before adding.
Question: √45 – √20
√45 = √(9 × 5) = √9 × √5 = 3 × √5 = 3√5
√20 = √(4 × 5) = √4 × √5 = 2 × √5 = 2√5
So √45 – √20 = 3√5 – 2√5 = √5
Answer: √5
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