Math Lesson 7.4.3 - Properties of Surds

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Welcome to our Math lesson on Properties of Surds, this is the third 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.

Properties of Surds

Surds have four main properties, where the first two are identical to the properties of roots we have discussed in tutorial 7.2. Let's list them below.

Surds property 1

For any positive numbers a and b (or they may also be negative when n is odd), we have

For example,

This rule is used in situations that require simplification such as those presented below.

Example 3

Write the simplest form of the expressions:

1. √21/√6
2. √105/√10 ∙ √3

Solution 3

1. From the first property of surds, we have
   √21/√6 = √(3 ∙ 7)/√(2 ∙ 3)
   = (√3 ∙ √7)/(√2 ∙ √3)
   = √7/√2
2. Again, from the first property of surds, we have
   √105/√10 ∙ √3 = √(3 ∙ 35)/√(2 ∙ 5) ∙ √3
   = √(3 ∙ 5 ∙ 7)/√(2 ∙ 5) ∙ √3
   = √3 ∙ √5 ∙ √7/√2 ∙ √5 ∙ √3
   = √7/√2

Surds property 2

For any positive numbers a and b (or they may also be negative when n is odd), we have

For example,

Example 4

Write the simplest form of the expressions:

1. √16/81
2. √88/33

Solution 4

1. From the second property of surds, we have
   ∜(16/81) = ∜16/∜81
   = ∜(2⁴ )/∜(3⁴ )
   = 2/3
2. Again, from second property of surds (combined with the first one), we have
   √(88/33) = √88/√33
   = √(8 ∙ 11)/√(3 ∙ 11)
   = √8 ∙ √11/√3 ∙ √11
   = √8/√3

Surds property 3

For any numbers a and b where b is positive, we have

Indeed,

For example,

This property is also known as "rationalisation of denominator", for which we will discuss extensively in the next tutorial.

Example 5

Write the following expressions in the simplest form.

1. 5/2√2
2. 12/√6

Solution 5

1. From the third property of surds (also in combination with the first two), we have
   5/2√2 = 5 ∙ √2/2√2 ∙ √2
   = 5 ∙ √2/2 ∙ (√2)²
   = 5 ∙ √2/2 ∙ 2
   = 5 ∙ √2/4
2. Again, from the third property of surds (also in combination with the first two), we have
   12/√6 = 12 ∙ √6/√6 ∙ √6
   = 12 ∙ √6/(√6)²
   = 12 ∙ √6/6
   = 2√6

Surds property 4

For any numbers a, b and positive c (or it can also be negative when n is odd), we have

This property derives from adding or subtracting with like term approach mentioned earlier in this tutorial.

For example,

Example 6

Write the following expressions in the simplest terms.

1. √112 - √28/√252
2. √50 + √18/5√8

Solution 6

1. From the fourth property of surds, in combination with the other three, we have
   (√112 - √28)/√252
   = √(16 ∙ 7) - √(4 ∙ 7)/√(36 ∙ 7)
   = √16 ∙ √7 - √4 ∙ √7/√36 ∙ √7
   = 4√7 - 2√7/6√7
   = √7 (4 - 2)/3 ∙ √7
   = 2√7/3√7
   = 2/3
2. Again, applying the fourth property of surds in combination with the other three, yields
   √50 + √18/5√8
   = √(25 ∙ 2) + √(9 ∙ 2)/5 ∙ √(4 ∙ 2)
   = √25 ∙ √2 + √9 ∙ √2/5 ∙ √4 ∙ √2
   = 5√2 + 3√2/5 ∙ 2√2
   = (5 + 3) √2/10√2
   = 8√2/10√2
   = 8/10
   = 4/5

More Surds Lessons and Learning Resources

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Enjoy the "Properties of Surds" math lesson? People who liked the "Surds lesson found the following resources useful:

1. Properties Feedback. Helps other - Leave a rating for this properties (see below)
2. Powers and Roots Math tutorial: Surds. Read the Surds math tutorial and build your math knowledge of Powers and Roots
3. 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
4. Powers and Roots Revision Notes: Surds. Print the notes so you can revise the key points covered in the math tutorial for Surds
5. Powers and Roots Practice Questions: Surds. Test and improve your knowledge of Surds with example questins and answers
6. Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
7. Continuing learning powers and roots - read our next math tutorial: Rationalising the Denominator

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