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Transcript
Simplifying Surds
Slideshow 6, Mr Richard Sasaki, Room 307
Objectives
β€’ Understand the meaning of rational
numbers
β€’ Understand the meaning of surd
β€’ Be able to check whether a number is a
surd or not
β€’ 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.
π‘₯ is rational if π‘₯ =
If π‘₯ =
𝑝
π‘ž
𝑝
π‘ž
where 𝑝, π‘ž ∈ β„€.
where 𝑝, π‘ž ∈ β„€, we say π‘₯ ∈ β„š. (π‘₯ is in the
rational number set, β„š.)
Rationality
If a number is not rational, we say that it
is irrational.
π‘₯ is irrational if it can’t be written in the form
𝑝
π‘ž where 𝑝, π‘ž ∈ β„€.
Therefore, an irrational number π‘₯ βˆ‰ β„š.
Example
Show that 0.8 ∈ β„š.
4
0.8 = 5 4, 5 ∈ β„€, ∴ 0.8 ∈ β„š.
Note: If π‘₯ =
𝑝
π‘ž,
π‘ž β‰  0.
Answers – Questions 1 - 4
0.45 =
9
20
and 9, 20 ∈ β„€.
βˆ’ 0.25 = βˆ“0.5 =
∴ 0.45 ∈ β„š.
where 1, 2 ∈ β„€. ∴
βˆ’ 0.25 ∈ β„š.
9 = ±3 where -3 and 3 are
integers. ∴ 9 ∈ β„€.
β„€
ℝ
β„š
ℝ
ℝ
β„€
1
βˆ“
2
Answers – Questions 5 - 6
Let π‘₯ = 0. 1 and 10π‘₯ = 1. 1.
∴ 9π‘₯ = 0. 1 βˆ’ 1. 1 = 1.
1
β‡’π‘₯=
9
Let π‘₯ = 0. 09 and
100π‘₯ = 9. 09.
∴ 99π‘₯ = 9. 09 βˆ’ 0. 09
= 9.
1
Let π‘₯ = 0. 5 and 10π‘₯ = 5. 5.
β‡’π‘₯=
11
∴ 9π‘₯ = 0. 5 βˆ’ 5. 5 = 5
5
β‡’π‘₯=
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, π‘Ž and 𝑏 where
π‘₯ = π‘Ž × π‘.
If we square both sides,2we get…
π‘₯2 = π‘Ž × π‘
π‘₯2 = π‘Ž × π‘ × π‘Ž × π‘
2
2
2
π‘₯ = π‘Ž × π‘
π‘₯2 = π‘Ž × π‘
If we square root both sides, we get…
π‘₯ = π‘Ž×𝑏
∴ π‘Ž × π‘ ≑ π‘Ž × π‘, where π‘Ž, 𝑏 ∈ ℝ.
Simplifying Surds
To simplify a surd, we need to write it in the
form π‘Ž 𝑏 where 𝑏 is as small as possible
and π‘Ž, 𝑏 ∈ β„€.
Note: Obviously, if π‘Ž = 1, π‘Ž 𝑏 = 𝑏.
Example
Simplify 8.
8 = 4βˆ™2 = 4βˆ™ 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
π‘Œπ‘’π‘ 
π‘π‘œ
π‘Œπ‘’π‘ 
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
π‘Ž 𝑏 where π‘Ž, 𝑏 ∈ β„€, 𝑏 >
1, 𝑏 βˆ‰ β„€.
16 = ±4 or rather 4 1 in
the form π‘Ž 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
β‘‘ 46
β‘‘ 23
14 2
39 3
1875
β‘’ 625
β‘€ 125
β‘€ 25
288
β‘‘ 144
β‘€ β‘€
β‘‘ 72
β‘‘ 36
1875 = 5
92 = 2 2 23
β‘‘ 18
= 25 3
= 2 23
β‘‘ 9
β‘’
β‘’
5
2
288 = 2
3 = 12 2
4
3