www.mathsrevision.com surds simplifying a surd rationalising a surd s4 credit

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SurdsSurds

Simplifying a Surd

Rationalising a Surd

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S4 Credit

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5. 2

Starter QuestionsStarter Questions

Use a calculator to find the values of :

1. 36 = 6

= 12

= 2

= 2

2. 144

33. 8 44. 16

1.41 2.7636. 21

S4 Credit

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www.mathsrevision.comwww.mathsrevision.comww

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1. To explain what a surd is and to investigate the rules for surds.

1.1. Learn rules for surds.Learn rules for surds.

The Laws Of Surds

1.1. Use rules to simplify surds.Use rules to simplify surds.

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SurdsSurds

N = {natural numbers}

= {1, 2, 3, 4, ……….}

W = {whole numbers}

= {0, 1, 2, 3, ………..}Z = {integers} = {….-2, -1, 0, 1, 2,

…..}Q = {rational numbers}

This is the set of all numbers which can be written as fractions or ratios.

eg 5 = 5/1 -7 = -7/1 0.6 = 6/10 = 3/5

55% = 55/100 = 11/20 etc

We can describe numbers by the following sets:

S4 Credit

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R = {real numbers}This is all possible numbers. If we plotted values on a number line then each of the previous sets would leave gaps but the set of real numbers would give us a solid line.

We should also note that

N “fits inside” W

W “fits inside” Z

Z “fits inside” Q

Q “fits inside” R

SurdsSurdsS4 Credit

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SurdsSurds

QZWN

When one set can fit inside another we say

that it is a subset of the other.

The members of R which are not inside Q are called irrational (Surd) numbers. These

cannot be expressed as fractions and include ,2, 35 etc

R

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2

What is a SurdWhat is a Surd

36 = 6

= 12

144

1.41 2.763 21

The above roots have exact values

and are called rational

These roots do NOT have exact values

and are called irrational OR Surds

S4 Credit

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Adding & Subtracting Surds

Adding and subtracting a surd such as 2. It can

be treated in the same way as an “x” variable in algebra. The following examples will illustrate this point.

4 2 + 6 2

=10 2

16 23 - 7 23

=9 23

10 3 + 7 3 - 4 3 =13 3

Note :

√2 + √3 does not equal √5S4 Credit

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First Rule

4 6 24

a b ab

4 10 40

List the first 10 square numbers

Examples

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

S4 Credit

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Simplifying Square Roots

Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea:

12

To simplify 12 we must split 12 into factors with at least one being a square number.

= 4 x 3

Now simplify the square root.

= 2 3

S4 Credit

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45 = 9 x 5= 35

32= 16 x 2= 42

72= 4 x 18

= 2 x 9 x 2= 2 x 3 x 2

= 62

Have a go !Think square numbersS4 Credit

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What Goes In The Box ?

Simplify the following square roots:

(1) 20 (2) 27 (3) 48

(4) 75 (5) 4500 (6) 3200

= 25

= 33

= 43

= 53

= 305 = 402

S4 Credit

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First Rule

4 4 29 39

a abb

25 25 5 1100 10 2100

Examples

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481

4

81

29

818

2 4

2 9

23

10500

10

10 50

1

50

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2 15

3

2 3 5

3

2 5

5 5

2 20

5 5

2 4 5

54

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19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept

Now try MIA

Ex 7.1 Ex 8.1 Ch9 (page 185)

S4 Credit

Exact Values

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Starter QuestionsStarter Questions

Simplify :

1. 20 = 2√5

= 3√2

= ¼

2. 18

1 13.

2 2

1 14.

4 4 =

¼

S4 Credit

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www.mathsrevision.comwww.mathsrevision.comww

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1. To explain how to rationalise a fractional surd.

1.1. Know that √a x √a = a.Know that √a x √a = a.

The Laws Of Surds

2.2. To be able to rationalise To be able to rationalise the numerator or the numerator or denominator of a denominator of a fractional surd.fractional surd.

S4 Credit

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Second Rule

4 4 4

a a a

13 13 13

Examples

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Rationalising Surds

You may recall from your fraction work that the top line of a fraction is the numerator and the bottom line the denominator.

2 numerator =

3 denominatorFractions can contain surds:

23

5

4 7

3 2

3 - 5

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Rationalising Surds

If by using certain maths techniques we remove the surd from either the top or bottom of the fraction then we say we are “rationalising the numerator” or “rationalising the denominator”.

Remember the rule a a a

This will help us to rationalise a surd fraction

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To rationalise the denominator multiply the top and bottom of the fraction by the square root you are

trying to remove:

3

53 5

=5 5

( 5 x 5 = 25 = 5 )

3 5=

5

Rationalising SurdsS4 Credit

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Let’s try this one :

Remember multiply top and bottom by root you are trying to remove

3

2 73 7

=2 7 7

3 7=

2 73 7

=14


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10

7 510 5

=7 5 5

10 5=

7 52 5

=7

Rationalising Surds

Rationalise the denominator

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What Goes In The Box ?

Rationalise the denominator of the following :

7

34

6

14

3 10

4

9 22 5

7 36 3

11 2

7 3=

32 6

=3

7 10=

15

2 29

2 15

=21

3 6=

11

S4 Credit

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Conjugate Pairs.

Rationalising Surds

Look at the expression : ( 5 2)( 5 2) This is a conjugate pair. The brackets are identical

apart from the sign in each bracket .

Multiplying out the brackets we get :

( 5 2)( 5 2) = 5 x 5 - 2 5 + 2 5 - 4

= 5 - 4

= 1When the brackets are multiplied out the surds ALWAYS cancel out and we end up seeing that the expression is rational ( no root sign )

S4 Credit

Looks something like the difference of two squares

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Third Rule

7 3 7 3

a b a b a b

Examples

11 5 11 5

Conjugate Pairs.

= 7 – 3 = 4

= 11 – 5 = 6

S4 Credit

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Rationalise the denominator in the expressions below by multiplying top and bottom by the

appropriate conjugate:

2

5 - 12( 5 + 1)

=( 5 - 1)( 5 + 1)

2( 5 + 1)=

( 5 5 - 5 + 5 - 1)2( 5 + 1)

=(5 - 1)

( 5 + 1)=

2

Conjugate Pairs.


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Rationalise the denominator in the expressions below by multiplying top and bottom by the

appropriate conjugate:

7

( 3 - 2)7( 3 + 2)

=( 3 - 2)( 3 + 2)

7( 3 + 2)=

(3 - 2)=7( 3 + 2)

Conjugate Pairs.


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What Goes In The Box

Rationalise the denominator in the expressions below :

5

( 7-2)3

( 3 - 2)

Rationalise the numerator in the expressions below :

6 + 412

5 + 117

= 3 + 6

- 5=6( 6 - 4)

- 6=7( 5 - 11)

5( 7 + 2)=

3

S4 Credit

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19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept

Now try MIA

Ex 9.1 Ex 9.1 Ch9 (page 188)

S4 Credit

Rationalising Surds

www.mathsrevision.com surds simplifying a surd rationalising a surd s4 credit

Documents

square numbers s4 credit

r surds s4 credit slide

rule examples s4 credit

r slide

s4 credit exact values

square numbers examples

rational numbers

irrational surd numbers