1. Surd Bracket Expansion
Slideshow 13 　Mathematics
Mr Richard Sasaki

2. Objectives Review bracket expansion types
Be able to square surd expressions in the form 𝑎 𝑏 +𝑐 and 𝑎 𝑏 +𝑐 𝑑
Multiply surd expressions in the form 𝑎 𝑏 +𝑐 and 𝑎 𝑏 +𝑐 𝑑

3. Review
We know how to multiply a pair of binomials. Let’s review the rules.
𝑥+𝑎 2 =
𝑥 2 +2𝑎𝑥+ 𝑎 2
𝑥+𝑎 𝑥−𝑎 =
𝑥 2 − 𝑎 2
𝑥−𝑎 2 =
𝑥 2 −2𝑎𝑥+ 𝑎 2
Let’s use these expressions to multiply surd expressions.

4. Surd Expressions
In chapter one, we multiplied surd expressions in the form 𝑎 𝑏 𝑐 𝑑 +𝑓 𝑔 .
𝑎 𝑏 𝑐 𝑑 +𝑓 𝑔 =
𝑎𝑐 𝑏𝑑 +𝑎𝑓 𝑔𝑏
Example
Expand and simplify 2 5 ( ). =
We know how to do these!

5. Surd Expressions
We also know how to conjugate surds. We used the rule 𝑥+𝑎 𝑥−𝑎 = 𝑥 2 − 𝑎 2 to produce two integers.
(𝑎 𝑏 +𝑐 𝑑 )(𝑎 𝑏 −𝑐 𝑑 )=
𝑎 2 𝑏− 𝑐 2 𝑑
Example
Write ( )(3 2 −6 11 ) as an integer.
− =
3 2 ∙2− 6 2 ∙11
=18−396
=−378
Note: Remember…
ℤ is the set. ℚ is the set.
integer
rational number

6. 37∈ ℤ +
53∈ ℤ +
291∈ ℤ +
98∈ ℤ +
148∈ ℤ +
−1396∈ ℤ −
= can’t be simplified further. ∴ ∉ℤ.
− = −6 81 = 18∈ℤ.
∴ − ∈ℤ.
4− = − =− 4 3 ∈ℚ as −4, 3 ∈ℤ.
∴ 4− ∈ℚ.

7. Squaring
Rules for surd expressions have the same rules as multiplying binomials.
𝑥+𝑎 2 =
𝑥 2 +2𝑎𝑥+ 𝑎 2
𝑎 𝑏 +𝑐 2 =
𝑎 2 𝑏+2𝑎𝑐 𝑏 + 𝑐 2
𝑎 𝑏 +𝑐 𝑑 2 =
𝑎 2 𝑏+2𝑎𝑐 𝑏𝑑 + 𝑐 2 𝑑
Example
Expand and simplify =
∙3 2 ∙4+ 4 2
=9∙ =
Note: All roots are positive as everything is squared.

8.
9−4 5
259−30 10
29−12 5
985−80 15
48−6 15
46−12 14 98
421−28 58

9. Multiplication Let’s review another rule… (𝑥+𝑎)(𝑦+𝑏)= 𝑥𝑦+𝑎𝑦+𝑥𝑏+𝑎𝑏
This is literally just gathering the combinations.
(𝑎 𝑏 +𝑐 𝑑 )(𝑚 𝑛 +𝑝 𝑞 )=
𝑎𝑚 𝑏𝑛 +𝑐𝑚 𝑑𝑛 +𝑎𝑝 𝑏𝑞 +𝑐𝑝 𝑑𝑞
Example
Expand and simplify 2 5 −
2 5 −
=2 5 ∙2 7 −3∙ ∙5−3∙5
=4 35 − −15

10.
3 14 − −8
48−16 3
40−22 5
− −1
−4 5 −10
22−6 5
117−113 3

11.
4 14 − −21
10 6 − −32
− −
−8 21 −36 7 − −210
− −
3616−
−57 6 −