Substituting Surds

Slideshow 18 　Mathematics

Mr Sasaki　　　 Room 307

Objectives

▪Substitute surd expressions into algebraic expressions▪Rationalise surd denominators when substituting

Substitution

As we have known for a long time now, substitution is the process of inserting expressions into other expressions.We’ve substituted numerical expressions and algebraic expressions. Today we will substitute surd expressions. The process is similar.

Surd SubstitutionIn a simple case, we just need to substitute and then simplify the expression.ExampleIf and , calculate .

𝑎2𝑏+2=¿(3 √2 )2 ∙2√3+2¿18 ∙2√3+2¿36 √3+2

Try the first worksheet!

Answers - Easy

2√14 6 √2 7± 4 4√14+4 8 √7−356 64 2√2

18 4√5+21 6 √10+3√2−1−4 √5−3 36 36 √5+18±6 √5±1020

Answers - Hard

√2+1 1 2√2+22√2 4 4√2+8√2+1 1 1

−4 6 √5+14−214 √5+30 6 √5+10 −64 √5−192√5+2√2+3 3√5+3−2√5−18

Other CasesLet’s recall a few properties about indices.

𝑥− 1≡1𝑥1

𝑥12

≡√𝑥𝑥

𝑥− 𝑦≡1𝑥 𝑦

ExampleIf , calculate .𝑥− 1=¿

1𝑥=¿

1√8

=¿12√2

=¿√2

2√2 ∙√2

¿ √24

Other CasesLet’s try a harder example.ExampleIf and , calculate . Only consider the positive root.

2𝑥−2𝑦 =¿ 2

𝑥2 𝑦=¿

2(√8+1)2 ∙3√2

=¿ 2(4√2+9)∙3√2

¿2

27√2+24¿ 227√2+24

∙ 27 √2−2427 √2−24¿

2 (27√2−24 )(27√2 )2−242

¿ 54 √2−481458−576

¿ 54√2−48882

¿ 9√2−8147

Answers - Easy√22 2√28

112

√22 √218

3−√27

√2+3 √2−13

4−6 √2 √77

Answers - Hard

√2+1 2√2+349

2√2+3 −17√2+7 8 √42+12√21111

13−4√3112√3 3+√11

11 13−4√3√1111 11√11