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×𝑏

If 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

1875

③ 625

⑤ 125

⑤ 25

⑤ ⑤

√1875=(√5 )4√3=25√3