Expanding Brackets with Surds and Fractions

Slideshow 9, Mr Richard Sasaki,Mathematics

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

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.

3 𝑥 (2 𝑥− 𝑦 )=¿6 𝑥2−3𝑥𝑦The same principles apply with surds.

2(√2−3)=¿2√2−6In this case, the expression cannot be simplified. But sometimes we are able to.

Expanding Brackets (Linear)Let’s try an example where we can simplify.

ExampleExpand and simplify .

4 (2√3+√12 )=¿8 √3+4 √12¿8 √3+4 ∙2∙√3¿8 √3+8√3

¿16 √3Note: We could simplify initially but then there would be no need to expand.

32√2 20√11 8 √3+40 √228+14√3 4√6+2√2 5√6−18 √26+2√3 3−√6 6 √5−510−√5 10+√15 7+2√7+√14−14−4√7 11−2√11 240−45√236 √70+18√10+12√2

Answers

Multiplying SurdsRemember, when we multiply a surd by itself, we will end up with two roots.

√3×√3¿±3But in actual fact, if we square a surd…it will always be positive.

(√3 )2¿3Can you see how these two things are different?Don’t forget to check whether the question requires positive roots or both roots too!

Note: If you say , this is acceptable.

Surds in FractionsWe had a look at some surd fractions in the form where . Let’s review.

ExampleSimplify .

12√3

=¿√3

2√3 ∙√3=¿√36

Remember, a fraction should have an integer as its denominator.

Surds in FractionsQuestions with different denominators require a different thought process. We need to expand brackets.ExampleSimplify .

4 √3+23

−2√7−54

=¿4 (4√3+2)3 ∙4

−3(2√7−5)4 ∙3

¿ 16√3+812

−6 √7−1512

¿ 16√3+8−6 √7+1512

¿ 16√3−6 √7+2312

√5+2√3+74

2√7−6√3+216

9√2−4 √7−4√5+2712

4 √6+96

23√5+36

7√3−5√7+7335

35√3−8 √628

95√3+42√26

Roots in DenominatorsCalculating with roots in denominators requires us to expand brackets where roots are on the outside.ExampleSimplify .

√2+1√3

− √3−1√2

=¿√3 (√2+1 )√3∙√3

−√2 (√3−1 )

√2 ∙√2¿ √6+√3

3− √6−√2

2¿2 (√6+√3 )

6−3 (√6−√2 )

6¿ 2√6+2√3

6−3√6−3 √2

6¿ 2√3+3√2−√6

6

11√5−96

27√1144

2√5+5√35

5√14+2√64

17√3−69√53

651√2−82√1035

4 √15+4√33

13√2−472

√3−6√2+123

4 √3+9√2−618

6√5+63 √2−16030

10√5−9√7+25415