Download Collecting Like Terms

Simplifying Surds
Slideshow 6, Mr Richard Sasaki, Room 307
Objectives
• Understand the meaning of rational
numbers
• Understand the meaning of surd
• Be able to check whether a number is a
surd or not
• Be able to simplify surds
Rationality
First we need to understand the meaning of
rational numbers.
What is a rational number?
A rational number is a number that can be
written in the form of a fraction.
𝑥 is rational if 𝑥 =
If 𝑥 =
𝑝
𝑞
𝑝
𝑞
where 𝑝, 𝑞 ∈ ℤ.
where 𝑝, 𝑞 ∈ ℤ, we say 𝑥 ∈ ℚ. (𝑥 is in the
rational number set, ℚ.)
Rationality
If a number is not rational, we say that it
is irrational.
𝑥 is irrational if it can’t be written in the form
𝑝
𝑞 where 𝑝, 𝑞 ∈ ℤ.
Therefore, an irrational number 𝑥 ∉ ℚ.
Example
Show that 0.8 ∈ ℚ.
4
0.8 = 5 4, 5 ∈ ℤ, ∴ 0.8 ∈ ℚ.
Note: If 𝑥 =
𝑝
𝑞,
𝑞 ≠ 0.
Answers – Questions 1 - 4
0.45 =
9
20
and 9, 20 ∈ ℤ.
− 0.25 = ∓0.5 =
∴ 0.45 ∈ ℚ.
where 1, 2 ∈ ℤ. ∴
− 0.25 ∈ ℚ.
9 = ±3 where -3 and 3 are
integers. ∴ 9 ∈ ℤ.
ℤ
ℝ
ℚ
ℝ
ℝ
ℤ
1
∓
2
Answers – Questions 5 - 6
Let 𝑥 = 0. 1 and 10𝑥 = 1. 1.
∴ 9𝑥 = 0. 1 − 1. 1 = 1.
1
⇒𝑥=
9
Let 𝑥 = 0. 09 and
100𝑥 = 9. 09.
∴ 99𝑥 = 9. 09 − 0. 09
= 9.
1
Let 𝑥 = 0. 5 and 10𝑥 = 5. 5.
⇒𝑥=
11
∴ 9𝑥 = 0. 5 − 5. 5 = 5
5
⇒𝑥=
9
Surds
What is a surd?
A surd is an irrational root of an integer. We
can’t remove its root symbol by simplifying it.
Are the following surds?
2
Yes!
2 5
30
No!
9
Yes!
Yes!
Even if the expression is not fully simplified,
if it is a root and irrational, it is a surd.
Multiplying Roots
How do we multiply square roots?
Let’s consider two roots, 𝑎 and 𝑏 where
𝑥 = 𝑎 × 𝑏.
If we square both sides,2we get…
𝑥2 = 𝑎 × 𝑏
𝑥2 = 𝑎 × 𝑏 × 𝑎 × 𝑏
2
2
2
𝑥 = 𝑎 × 𝑏
𝑥2 = 𝑎 × 𝑏
If we square root both sides, we get…
𝑥 = 𝑎×𝑏
∴ 𝑎 × 𝑏 ≡ 𝑎 × 𝑏, where 𝑎, 𝑏 ∈ ℝ.
Simplifying Surds
To simplify a surd, we need to write it in the
form 𝑎 𝑏 where 𝑏 is as small as possible
and 𝑎, 𝑏 ∈ ℤ.
Note: Obviously, if 𝑎 = 1, 𝑎 𝑏 = 𝑏.
Example
Simplify 8.
8 = 4∙2 = 4∙ 2 =2 2
We try to take remove square factors out and
simplify them by removing their square root
symbol.
Answers - Easy
Because 2 has positive
and negative roots anyway.
∴ 2 2 ≡ ±2 2.
4 2
2 5
10 3
6 2
5 6
4 6
𝑌𝑒𝑠
𝑁𝑜
𝑌𝑒𝑠
6 3
±6
42 2 12 6
16 2
21 6
No, of course not! 6 is a
surd but 6 is not prime.
Square numbers.
Answers – Hard (Questions 1 – 3)
Let 16 be in the form
𝑎 𝑏 where 𝑎, 𝑏 ∈ ℤ, 𝑏 >
1, 𝑏 ∉ ℤ.
16 = ±4 or rather 4 1 in
the form 𝑎 𝑏. As 𝑏 = 1,
16 is not a surd.
8 3
2 93
9 10
24 3 17 5
±22
Answers – Hard (Questions 4 – 5)
10 15
92
② 46
② 23
14 2
39 3
1875
③ 625
⑤ 125
⑤ 25
288
② 144
⑤ ⑤
② 72
② 36
1875 = 5
92 = 2 2 23
② 18
= 25 3
= 2 23
② 9
③
③
5
2
288 = 2
3 = 12 2
4
3