Expanding Brackets with Surds and Fractions
Slideshow 9, Mr Richard Sasaki,Mathematics
Objectivesβ’ Be able to expand brackets with surdsβ’ Expanding brackets with surds on the outsideβ’ Calculate with surds in fractions
Expanding Brackets (Linear)Letβs think back to algebra.
When we expand brackets, we multiply terms on the inside by the one on the outside.
3 π₯ (2 π₯β π¦ )=ΒΏ6 π₯2β3π₯π¦The same principles apply with surds.
2(β2β3)=ΒΏ2β2β6In this case, the expression cannot be simplified. But sometimes we are able to.
Expanding Brackets (Linear)Letβs try an example where we can simplify.
ExampleExpand and simplify .
4 (2β3+β12 )=ΒΏ8 β3+4 β12ΒΏ8 β3+4 β2ββ3ΒΏ8 β3+8β3
ΒΏ16 β3Note: We could simplify initially but then there would be no need to expand.
32β2 20β11 8 β3+40 β228+14β3 4β6+2β2 5β6β18 β26+2β3 3ββ6 6 β5β510ββ5 10+β15 7+2β7+β14β14β4β7 11β2β11 240β45β236 β70+18β10+12β2
Answers
Multiplying SurdsRemember, when we multiply a surd by itself, we will end up with two roots.
β3Γβ3ΒΏΒ±3But in actual fact, if we square a surdβ¦it will always be positive.
(β3 )2ΒΏ3Can you see how these two things are different?Donβt 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 . Letβs review.
ExampleSimplify .
12β3
=ΒΏβ3
2β3 ββ3=ΒΏβ36
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 β3+23
β2β7β54
=ΒΏ4 (4β3+2)3 β4
β3(2β7β5)4 β3
ΒΏ 16β3+812
β6 β7β1512
ΒΏ 16β3+8β6 β7+1512
ΒΏ 16β3β6 β7+2312
β5+2β3+74
2β7β6β3+216
9β2β4 β7β4β5+2712
4 β6+96
23β5+36
7β3β5β7+7335
35β3β8 β628
95β3+42β26
Roots in DenominatorsCalculating with roots in denominators requires us to expand brackets where roots are on the outside.ExampleSimplify .
β2+1β3
β β3β1β2
=ΒΏβ3 (β2+1 )β3ββ3
ββ2 (β3β1 )
β2 ββ2ΒΏ β6+β3
3β β6ββ2
2ΒΏ2 (β6+β3 )
6β3 (β6ββ2 )
6ΒΏ 2β6+2β3
6β3β6β3 β2
6ΒΏ 2β3+3β2ββ6
6
11β5β96
27β1144
2β5+5β35
5β14+2β64
17β3β69β53
651β2β82β1035
4 β15+4β33
13β2β472
β3β6β2+123
4 β3+9β2β618
6β5+63 β2β16030
10β5β9β7+25415
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