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Surds & Surds & IndicesIndices
Simplifying a SurdRationalising a Surd
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Conjugate Pairs (EXTENSION)
Nat 5
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
Nat 5
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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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ab
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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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What is a Surd ?What is a Surd ?
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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What is a Surd ?What is a Surd ?
Find the exact value of sinxo.
Nat 5
√21
xo
O
HSin xo =
Sin xo = 1
√2
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Now try N5 TJ
Ex 17.1
Ch17 (page 170)
Nat 5
What is a Surd ?What is a Surd ?
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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.
Nat 5
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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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21 Apr 202321 Apr 2023
Problem : Find the length of space diagonal AG.First find AH2 :
Nat 5
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)
Nat 5
Surds
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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
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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.
Nat 5
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Second Rule
4 4 4
a a a
13 13 13
Examples
Nat 5
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Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. Lafferty@mathsrevision.comLafferty@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 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. Lafferty@mathsrevision.comLafferty@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
Rationalising SurdsNat 5
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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
Nat 5
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Now try N5 TJ
Ex 17.2 Q8 ... Q10
Ch17 (page 172)
Nat 5
Surds
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3. 12 + 3 12 - 3
Starter QuestionsStarter Questions
Multiply out :
1. 3 3 = 3
= 14
2. 14 14
= 12- 9 = 3
Conjugate Pairs.Nat 5
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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
Nat 5
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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.
Rationalising SurdsNat 5
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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.
Rationalising SurdsNat 5
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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
Nat 5
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Now try N5 TJ
Ex 17.2 Q8 ... Q10
Ch17 (page 172)
Nat 5
Surds
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Starter QuestionsStarter Questions
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
Nat 5
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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.
Nat 5
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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
Nat 5
= 32
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IndicesIndices
Write down 5 x 5 x 5 x 5 in indices format.
Nat 5
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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What Goes In The Box ?
103
(-2)8
-(2)8
90
1000
256
Nat 5
Use your calculator to work out the following
-256
1
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Now try N5 TJ
Ex 17.3
Ch17 (page 173)
Nat 5
Indices
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Starter QuestionsStarter Questions
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
Can you spot the connection !
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Rule 2
am ÷ an = a(m - n) simply subtract
powers
Calculate : 92 = 81
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What Goes In The Box ?
3a
b3 x b5 =
y9 ÷ y5
=
f4 x g5 =
a3 x a0 =
b8
y4
4 5f g
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What Goes In The Box ?
q3 x q4
3y4 x 5y5
e5 x e3 x e-
6
3p8 x 2p2 x 5p-3
q7
15y9
Nat 5
Simplify the following using indices rules
e2
30p7
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What Goes In The Box ?
q3
3d5
Nat 5
Simplify the following using indices rules
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)
Nat 5
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
Can you spot the connection !
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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What Goes In The Box ?
(b3)0
(y0)-2
(c-3)4
1
1
Nat 5
c-12
(3d2)2
9d4
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Now try N5 TJ
Ex 17.4 Q7 ... Q13
Ch17 (page 175)
Nat 5
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
Fractions as IndicesFractions as IndicesNat 5
Rule 5
a-m = 1am
1am
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What Goes In The Box ?
u-4
(w4)-2
Nat 5
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)
Nat 5
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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Fractions as IndicesFractions as IndicesNat 5
7 4x4
7x
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Fractions as IndicesFractions as IndicesNat 5
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
Nat 5
1 32 24
a
322
a
434 m 3 44 m
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Fractions as IndicesFractions as Indices
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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Fractions as IndicesFractions as Indices
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)
Nat 5
Indices