Math Lesson 7.4.2 - What are the different Types of Surds

Welcome to our Math lesson on What are the different Types of Surds, this is the second lesson of our suite of math lessons covering the topic of Surds, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Types of Surds

Surds are classified into types for a better understanding. It is worth noting here that a surd may belong to more than one category at the same time. Surds may belong to one (or more) of the following categories:

1. Simple surds. This category includes surds that contain only one number inside the root.
For example, √2; √3; √10; ∛21; 40, etc., are all examples of simple surds as they contain only one number inside the root.
2. Pure surds. This category includes only surds obtained when taking the root of a prime number.
For example, √2; √3; √5; √23; 41, etc., are all examples of pure surds as all numbers inside the roots are prime (i.e. divisible only by 1 and themselves).
3. Similar surds. This category includes surds that have common factors. In other words, similar surds must contain the same number inside the root after some operations despite initially they may seem as unrelated.
For example, √8 and √18 are similar surds because from the properties of roots we have
√8 = √(4 ∙ 2) = √4 ∙ √2 = 2 ∙ √2
and
√18 = √(9 ∙ 2) = √9 ∙ √2 = 3 ∙ √2
Hence, since both numbers has the same irrational number (√2) as a common factor, they represent similar surds.
4. Mixed surds. This category includes surds that contain numbers that can be expressed as products of rational and irrational numbers, such as those shown in the previous category. Some other examples of numbers that can be expressed as mixed surds include √27; √50, etc., because
√27 = √(9 ∙ 3) = √9 ∙ √3 = 3 ∙ √3
and
√50 = √(25 ∙ 2) = √25 ∙ √2 = 5 ∙ √2
5. Compound surds. This category includes the sum or difference of two or more individual surds.
For example, √2 + √3; √7 - ∛21, etc., are compound surds.
6. Binomial surds. This category includes two or more surds that when added or subtracted give a single surd. Obviously, the original surds must be similar, so that the operations with them be possible. We have encountered a similar situation when adding or subtracting like terms in algebraic expressions.
For example, √2 + √8 is an example of binomial surds because
√2 + √8 = √2 + √(4 ∙ 2)
= √2 + √4 ∙ √2
= √2 + 2 ∙ √2
= 1 ∙ √2 + 2 ∙ √2
= (1 + 2) ∙ √2
= 3√2

Example 2

What kind of surds are those given below at the after making all possible operations?

1. ∛128
2. √12 - √27
3. √32 + √7

Solution 2

1. In the original form, the first number represents a simple surd as it contains a single number inside the root. On the other hand, when doing the operations we obtain a mixed surd, as
∛128 = ∛(64 ∙ 2)
= ∛64 ∙ ∛2
= ∛(43 ) ∙ ∛2
= 4∛2
2. In the original form, we have a compound surd, as there is an addition of two individual surds. After making the operations, we first obtain first a binomial surd, and eventually a mixed surd, as
√12 - √27 = √(4 ∙ 3) - √(9 ∙ 3)
= √4 ∙ √3-√9 ∙ √3
= 2√3 - 3√3
= -1√3
= -√3
3. This is an example of compound surds as the components surds are not similar, so that to allow us make operations with them. The only thing we can do is to convert the first component surd (which is a simple surd) into a mixed surd, i.e.
√32 + √7 = √(16 ∙ 2) + √7
= √16 ∙ √2 + √7
= 4√2 + √7

More Surds Lessons and Learning Resources

Powers and Roots Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
7.4Surds
Lesson IDMath Lesson TitleLessonVideo
Lesson
7.4.1What Are Surds?
7.4.2What are the different Types of Surds
7.4.3Properties of Surds
7.4.4Basic Operations with Surds

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7. Continuing learning powers and roots - read our next math tutorial: Rationalising the Denominator