1. Slideshow 10, Mr Richard Sasaki, Mathematics
Conjugations in Surds
Slideshow 10, Mr Richard Sasaki, Mathematics

2. Objectives
Be able to rationalise denominators of fractions in the form 𝑎 +𝑏
Do the same for denominators in the form 𝑎 𝑏 +𝑐
Simplify expressions with surds in this form

3. Rationalising the Denominator
Previously, we learned how to make the denominator an integer on a fraction.
1∙ ∙ 2 =
2 2 =
If the denominator is in the form 𝑎 𝑏 , we can multiply the top and bottom by 𝑏 as 𝑎 𝑏 ∙ 𝑏 =𝑎𝑏.
How about denominators in the form 𝑎 +𝑏?
We need to make a conjugation. What’s that?

4. Conjugation
In maths (not English), a conjugation refers to things that link to one another (a relationship). The conjugate of 𝑎 +𝑏 is .
𝑎 −𝑏
Example
Simplify =
∙ 2 −1 2 −1 =
2 − − 1 2
= 2 −1 2−1 =
2 −1

5. Conjugation Example Simplify 3 2 5 −2 . 3 2 5 −2 =
−2 =
−2 ∙ =
3 2 ∙( 5 +2) − 2 2
= −4 =

6. 2 −3
5 +2
3 +1
3 −1 2
5 +2
−2 2 −3 −
3 +3 4
−5 3 −3 11
−54 7 −42 37
6 6 −12
−7 10 −
−18 223
− −600 19

7. Answers – Part 1, Hard 12−6 2 4 2 −2 6 5 +2 30−5 2 17 3− 3 2 18 2 −18
12−6 2
4 2 −2 6
5 +2
30−
3− 3 2
18 2 −18 4 10 21
−6 5
4 15 −
− 2 +2

8. Other Types
When we need to conjugate a denominator in the form 𝑎 𝑏 +𝑐, we multiply the numerator and denominator by
𝑎 𝑏 −𝑐
Example
Simplify −5 .
−5 =
−5 ∙ =
= 3 5 ∙( ) − 5 2
= −25
= −6 15 −

9. Answers – Part 2, Easy 3 2 +1 9 3 −5 5 7 −2 3 2 +4 2 5 3 −2 71
9 3 −5
5 7 −2
5 3 −2 71 −
− 6 −
18−
−2 7 −1 27

10. 3 + 2
7 − 5
2 3 −3 2
4 2 −
3 42 −14 13
3 − 2
6 +3
− −