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
Rational Numbers
What is a rational number?
A rational number is a number that can be
written in the form of a fraction where the
numerator and denominator are integers.
𝑝
𝑞
𝑥 is rational if 𝑥 = where 𝑝, 𝑞 ∈ ℤ.
𝑝
𝑞
If 𝑥 = 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. As 4, 5 ∈ ℤ, ∴ 0.8 ∈ ℚ.
𝑝
Note: If 𝑥 = 𝑞, 𝑞 ≠ 0 (You can’t divide a
number by 0.)
0.45 =
9
20
and 9, 20 ∈ ℤ. ∴ 0.45 ∈ ℚ.
9 = ±3 where -3 and 3 are integers. ∴ 9 ∈ ℤ.
ℤ
ℚ
− 0.25 = ∓0.5 =
ℝ
1
∓
2
ℝ
ℝ
ℤ
where 1, 2 ∈ ℤ. ∴ − 0.25 ∈ ℚ.
𝑥 = 0. 1, 10𝑥 = 1. 1.
∴ 9𝑥 = 1. 1 − 0. 1 = 1.
⇒𝑥=1 9
𝑥 = 0. 09, 100𝑥 = 9. 09.
∴ 99𝑥 = 9. 09 − 0. 09 = 9.
⇒ 𝑥 = 1 11
𝑥 = 0. 5, 10𝑥 = 5. 5.
∴ 9𝑥 = 5. 5 − 0. 5 = 5.
⇒𝑥=5 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
9
Yes!
No!
2 5
32
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 surds and roots?
Let’s consider two roots, 𝑎 and 𝑏 where
𝑥 = 𝑎 × 𝑏.
If we square both sides, we
get…
2
2
𝑥 = 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
Yes
6 3
10 3
2 5
No
6
16 2
6 2
5 6
4 6
Yes
42 2
12 6
No, for example, 6 is a surd but 6 is not prime.
Square numbers.
21 6
Answers – Hard (Questions
1 – 4)
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
9 10
2 93
24 3
±22
17 5
ℤ
10 15
14 2
39 3
Answers – Hard (Question 5)
288
1875
625
② 144
③
② 72
⑤ 125
36
②
⑤ 25
92 = 2 2 23
= 2 23
② 18
⑤⑤
4
② 9
1875 = 5
3
5
2
288 = 2
3 ③ ③
= 25 3
= 12 2
92
② 46
② 23