Www.mathsrevision.com Surds & Indices Simplifying a Surd Rationalising a Surd Conjugate Pairs...

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Surds & Surds & IndicesIndices

Simplifying a SurdRationalising a Surd

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Conjugate Pairs (EXTENSION)

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Add/Sub Indices

Power of a Power

Negative / Positive Indices

Fraction Indices

Exam Type Questions

What is a surd ? What are Indices

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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

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1. We are learning what a surd is and why it is used.

1.1. Understand what a surds Understand what a surds is.is.

What is a Surds ?

2. 2. Recognise questions that Recognise questions that may contain surds.may contain surds.

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2

What is a What is a Surd ?Surd ?

36 = 6

= 12

144

1.41..... 2.76.....3 21

The above roots have exact values

and are called rational

These roots CANNOT be written in the form

and are called irrational root OR Surds

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What is a Surd ?What is a Surd ?

Which of the following are surds.

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81 3 64 8

x2 = 72 + 12

x2 = 50

x = √50

√

x = 5√2

x = √25 √2

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Solve the equation leaving you answers in surd format :

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2x2 + 7 = 11-7 -7

2x2 = 4÷2

x2 = 2

√ x = ±√2

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Find the exact value of sinxo.

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√21

xo

O

HSin xo =

Sin xo = 1

√2

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Now try N5 TJ

Ex 17.1

Ch17 (page 170)

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1. We are learning rules for simplify surds.

1.1. Understand the basic rules Understand the basic rules for surds.for surds.

Simplifying Surds

2. 2. Use rules to simplify surds.Use rules to simplify surds.

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Adding & Subtracting Surds

We can only adding and subtracting a surds that have the same surd. It can be treated in the same way as “like terms” 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 √5

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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

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Simplifying Surds

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

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All to do with

Square numbers.

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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 numbersNat 5

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What Goes In The Box ?

Simplify the following square roots:

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

(4) 3 x 8 (5) 6 x 12 (6) 3 x 5 x 15

= 25

= 33

= 43

= 26

= 62 = 15

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18 Apr 202318 Apr 2023

Problem : Find the length of space diagonal AG.First find AH2 :

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A

B C

D

E

F G

H

10cm

10cm

10cm

10cm

3D Pythagoras 3D Pythagoras TheoremTheorem

222 )()()( DHADAH

222 )()()( HGAHAG

200)( 2 AH

222 )10()10()( AH

300)10(200)( 22 AG

cm 3103100300 AG

Next AG :

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Now try N5 TJ

Ex 17.2 Q1 ... Q7

Ch17 (page 171)

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Surds

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Simplify :

1. 20 = 2√5

= 3√2

= ¼

2. 18

1 13.

2 2

1 14.

4 4 =

¼

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1. We are learning how to multiply out a bracket containing surds and how to rationalise a fractional surd.

1.1. Know that √a x √b = √abKnow that √a x √b = √ab

The Laws Of Surds

2.2. Use multiplication table to Use multiplication table to simplify surds in brackets.simplify surds in brackets.

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

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Second Rule

4 4 4

a a a

13 13 13

Examples

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Apr 18, 2023Apr 18, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Surds with Surds with BracketsBrackets

(√6 + 3)(√6 + 5) √6+ 3 + 5

5√6

+153√6

6

21+ 8√6

Example

Tidy up !

√6

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Apr 18, 2023Apr 18, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Surds with Surds with BracketsBrackets

(√2 + 4)(√2 + 4) √2+ 4 + 4

4√2

+164√2

2

18+ 8√2

Example

Tidy up !

√2

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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 SurdsNat 5

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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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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

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Now try N5 TJ

Ex 17.2 Q8 ... Q10

Ch17 (page 172)

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Surds

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3. 12 + 3 12 - 3


Multiply out :

1. 3 3 = 3

= 14

2. 14 14

= 12- 9 = 3

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1. To explain how to use the conjugate pair to rationalise a complex fractional surd.

1.1. Know that Know that (√a + √b)(√a + √b)(√a - √b) (√a - √b) = a - b= a - b

The Laws Of Surds

2.2. To be able to use the To be able to use the conjugate pair to conjugate pair to rationalise complex rationalise complex fractional surd.fractional surd.

Conjugate Pairs.Nat 5

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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 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 )

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

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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

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Now try N5 TJ

Ex 17.2 Q8 ... Q10

Ch17 (page 172)

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Surds

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1. Simplify the following fractions :

7 7 a a (a) (b)

b b 2a 2d

2. Simplif y 2c(4 - c) - 5(4+c)

3. Multiply out (x +1)(x -5)

4. Simplif y 2 27 -5 3

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1. We are learning what indices are and how to use our calculator to deal with calculations containing indices.

1.1. Understand what indices Understand what indices are.are.

Indices

2.2. Be able you calculator to Be able you calculator to do calculations containing do calculations containing indices.indices.

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IndicesIndices

an is a short hand way of writing

a x a x a ……. (n factors)

a is called the base number

and n is called the index number

Calculate :

2 x 2 x 2 x 2 x 2

Calculate : 25 = 32

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= 32

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IndicesIndices

Write down 5 x 5 x 5 x 5 in indices format.

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54

Find the value of the index for each below

3x = 27

x = 3

2x = 64

x = 6

12x = 144

x = 2

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103

(-2)8

-(2)8

90

1000

256

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Use your calculator to work out the following

-256

1

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Now try N5 TJ

Ex 17.3

Ch17 (page 173)

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Indices

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1. Simplify the following fractions :

3

u 5 a a (a) (b)

10 u 2a 2d

22. Factorise 3x 9x

23. Factorise x +3x +2

4. Simplif y 10 27 5 3

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1. We are learning various rules for indices.

1.1. Understand basic rules for Understand basic rules for indices.indices.

Indices

2.2. Use rules to simplify Use rules to simplify indices.indices.

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IndicesIndices

Calculate : 43 x 42 = 1024

Can you spot the connection !

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Rule 1

am x an = a(m + n)

simply add powers

Calculate : 45 = 1024

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IndicesIndices

Calculate : 95 ÷ 93 = 81


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Rule 2

am ÷ an = a(m - n) simply subtract

powers

Calculate : 92 = 81

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3a

b3 x b5 =

y9 ÷ y5

=

f4 x g5 =

a3 x a0 =

b8

y4

4 5f g

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q3 x q4

3y4 x 5y5

e5 x e3 x e-

6

3p8 x 2p2 x 5p-3

q7

15y9

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Simplify the following using indices rules

e2

30p7

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q3

3d5

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e-2

q9

q6

e6

e8

6d8

2d3

15g3h7

3g5h5

5h2

g2

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Now try N5 TJ

Ex 17.4 Q1 ... Q6

Ch17 (page 174)

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Indices

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Another Rule

35a 5 5 5= a a a 5 + 5 + 5 15=a a

53a 3 3 3 3 3= a a a a a 3 + 3 + 3 + 3 + 3 15=a a

Power of a PowerPower of a PowerNat 5


Rule 3

(am)n = amn simply multiply powers

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More Rules5

5

aa

a a a a a=a a a a a

= 1

5

5

aa

5 - 5= a 0= a

Fractions as IndicesFractions as IndicesNat 5

Rule 4

a0 = 1

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(b3)0

(y0)-2

(c-3)4

1

1

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c-12

(3d2)2

9d4

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q3 x q4

3y4 x 5y5

e5 x e3 x e-

6

3p8 x 2p2 x 5p-3

q7

15y9

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e2

30p7

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Now try N5 TJ

Ex 17.4 Q7 ... Q13

Ch17 (page 175)

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Indices

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More Rules3

5

aa

a a a=a a a a a

2

1=a

3 - 5= a

By the division rule3

5

aa

-2= a


Rule 5

a-m = 1am

1am

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u-4

(w4)-2

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Write as a positive power

y31u4

1y-3

1w8

h6

h10( (-2

h8

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Now try N5 TJ

Ex 17.4 Q14 onwards

Ch17 (page 176)

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Indices

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1. To show how to simplify harder fractional indices.

1.1. Simplify harder fractional Simplify harder fractional indices.indices.

Algebraic OperationsNat 5

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7 4x4

7x

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Rule 6

nn m nm ma = a = a

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Fractions as IndicesFractions as Indices

Example : Change to index form

-34 a

1

4 364 m1 43 364 m

1

-3 24 a

Example : Change to surd form

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1 32 24

a

322

a

434 m 3 44 m

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n

n m nm m a = a = a Examples

834 y

244= y 6= y

34 16 3

4= 16 3= (2) = 8

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n

n m nm m a = a = a Examples

53 27

53

1=

27 53 27

1=

531

= 2431

=

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Now try N5 TJ

Ex 17.5

Ch17 (page 177)

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