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
RationalityFirst 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.
is rational if where .
If where , we say . ( is in the rational number set, .)
RationalityIf a number is not rational, we say that it is .irration
al is irrational if it can’t be written in the form where .Therefore, an irrational number .ExampleShow that 0.8 .
0.8=¿45 .Note: If , .
Answers – Questions 1 - 4 and ..
where -3 and 3 are integers. .
ℤ ℚ ℝ
ℝ ℝ ℤ
where . .
Answers – Questions 5 - 6
Let and
Let and
Let and
SurdsWhat 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!√9 No!
2√5 Yes!√30 Yes!
Even if the expression is not fully simplified, if it is a root and irrational, it is a surd.
Multiplying RootsHow do we multiply square roots?Let’s consider two roots, and where .
If we square both sides, we get… 𝑥2=(√𝑎×√𝑏)2
𝑥2=√𝑎×√𝑏×√𝑎×√𝑏𝑥2=(√𝑎)2× (√𝑏)2𝑥2=𝑎×𝑏If we square root both sides, we
get… 𝑥=√𝑎×𝑏, where .
Simplifying SurdsTo simplify a surd, we need to write it in the form where is as small as possible and .
Note: Obviously, if .ExampleSimplify .
√8=¿ √ 4 ∙√2=¿2√2We try to take remove square factors out and simplify them by removing their square root symbol.
√ 4 ∙2=¿
Answers - EasyBecause has positive and negative roots anyway. .
4√2 2√5 10√36 √2 5√6 4√6
𝑌𝑒𝑠 𝑁𝑜 𝑌𝑒𝑠
6 √3±6 16√242√212√6 21√6
No, of course not! is a surd but 6 is not prime.
Square numbers.
Answers – Hard (Questions 1 – 3)
Let be in the form where . or rather in the form . As , is not a surd.
8 √3 2√93±229√10 24 √317√5
ℤ
Answers – Hard (Questions 4 – 5)
10√15 14 √2 39√3
92② 46
② 23
√92=√2√2√23¿2√23
288② 144
② 72② 36
②18②
9③ ③
√288=(√2 )5 (√3 )2=12√2
1875③ 625⑤ 125
⑤ 25⑤ ⑤
√1875=(√5 )4√3=25√3
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