Math Lesson 7.4.4 - Basic Operations with Surds

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Welcome to our Math lesson on Basic Operations with Surds, this is the fourth 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.

Basic Operations with Surds

We can make operations with surds in the same way as with all the other types of numbers. Let's explain them briefly.

Addition and subtraction of surds

We can add or subtract only like surds. Some surds apparently look different, but after a few operations, we can convert them into like surds. Let's consider an example to clarify this point.

Example 7

Simplify the following expressions containing surds.

√12 + √27
3√20 - √45

Solution 7

First, we must break down the original surds into factors to see whether they contain like surds or not. If yes, we do the operations by collecting like surds. We have
√12 + √27
= √(4 ∙ 3) + √9 ∙ 3
= √4 ∙ √3 + √9 ∙ √3
= 2√3 + 5 ∙ 3√3
= 2√3 + 15√3
= 17√3
Again, we use the same procedure as in (a). Thus,
3√20-√45
= 3√(4 ∙ 5) - √(9 ∙ 5)
= 3 ∙ √4 ∙ √5 - √9 ∙ √5
= 3 ∙ 2 ∙ √5 - 3 ∙ √5
= 6√5 - 3√5
= 3√5

Multiplication and division of surds

Multiplication and division of surds is nothing more but the applications of the first two properties of surds in the reverse direction (from end to start). The only restriction is that surds must have the same index. Mathematically, we have:

√a ∙ √b = √a ∙ b

This operation is particularly useful when trying to take numbers out of roots. For example, in the expression

∛4 ∙ ∛2

we can't do any operation in the actual condition. Hence, we write

∛4 ∙ ∛2 = ∛(4 ∙ 2)
= ∛8
= ∛(2³ )
= 2

We can use the same approach in division of surds as well. We can write the second property of surds in the reverse direction, i.e.

√a/√b = √a/b

For example,

√(3/4) = √3/√4 = √3/2

This method too, is used to solve or simplify an expression where it is not possible to make any operation or simplification of surds in the original form.

Example 8

Write the following expressions in the simplest form.

√320/√20
√132 ∙ √7/√77

Solution 8

We have

√320/√20 = √(320/20)
= √16
= 4

We have

√132 ∙ √7/√77 = √132 ∙ √7/√77
= √132 ∙ √(7/77)
= √132 ∙ √(1/11)
= √132 ∙ √1/√11
= √132/√11
= √(12 ∙ 11)/√11
= (√12 ∙ √11)/√11
= √12
= √(4 ∙ 3)
= √4 ∙ √3
= 2√3

More Surds Lessons and Learning Resources

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

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Powers and Roots Math tutorial: Surds. Read the Surds math tutorial and build your math knowledge of Powers and Roots
Powers and Roots Video tutorial: Surds. Watch or listen to the Surds video tutorial, a useful way to help you revise when travelling to and from school/college
Powers and Roots Revision Notes: Surds. Print the notes so you can revise the key points covered in the math tutorial for Surds
Powers and Roots Practice Questions: Surds. Test and improve your knowledge of Surds with example questins and answers
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Continuing learning powers and roots - read our next math tutorial: Rationalising the Denominator

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