1. Expanding Brackets with Surds and Fractions
Slideshow 9, Mr Richard Sasaki,
Mathematics

2. Objectives Be able to expand brackets with surds
Expanding brackets with surds on the outside
Calculate with surds in fractions

3. Expanding Brackets (Linear)
Let’s think back to algebra.
When we expand brackets, we multiply terms on the inside by the one on the outside.
3𝑥 2𝑥−𝑦 =
6 𝑥 2 −3𝑥𝑦
The same principles apply with surds.
2( 2 −3)=
2 2 −6
In this case, the expression cannot be simplified. But sometimes we are able to.

4. Expanding Brackets (Linear)
Let’s try an example where we can simplify.
Example
Expand and simplify 4( ). =
= ∙2∙ 3 =
=16 3
Note: We could simplify initially but then there would be no need to expand.

5. 32 2
20 11
5 6 −18 2
6+2 3
3− 6
6 5 −5
10− 5
−14−4 7
11−2 11
240−45 2

6. Surds in Fractions
We had a look at some surd fractions in the form 𝑎 𝑏 𝑐 𝑑 where 𝑎, 𝑏, 𝑐, 𝑑∈ℤ (𝑐, 𝑑≠0). Let’s review.
Example
Simplify
∙ 3 =
3 6 =
Remember, a fraction should have an integer as its denominator.

7. Surds in Fractions
Questions with different denominators require a different thought process. We need to expand brackets.
Example
Simplify − 2 7 −5 4 .
− 2 7 −5 4 =
4( ) 3∙4 − 3(2 7 −5) 4∙3
= − 6 7 −15 12
= −
= −

8.
2 7 −
9 2 −4 7 −
7 3 −
35 3 −

9. Roots in Denominators
Calculating with roots in denominators requires us to expand brackets where roots are on the outside.
Example
Simplify − 3 −
− 3 − =
∙ 3 − − ∙ 2
= − 6 − 2 2
= − −
= − 3 6 −
= − 6 6

10. 11 5 −9 6
17 3 − −

11.
13 2 −4 72
3 −
−6 18
−160 30
10 5 −