SCHOLAR Study Guide National 5 Mathematics …...TOPIC 1. SURDS AND INDICES 3 1.1 Simplifying surds...

SCHOLAR Study Guide

National 5 Mathematics

Course MaterialsTopic 22: Surds and indices

Authored by:Margaret Ferguson

Reviewed by:Jillian Hornby

Previously authored by:Eddie Mullan

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2016 by Heriot-Watt University SCHOLAR.

Copyright © 2016 SCHOLAR Forum.

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Distributed by the SCHOLAR Forum.

SCHOLAR Study Guide Course Materials Topic 22: National 5 Mathematics

1. National 5 Mathematics Course Code: C747 75

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1

Topic 1

Surds and indices

Contents

22.1 Simplifying surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

22.2 Collecting like terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

22.3 Rationalising denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

22.4 Multiplication and division of terms with positive indices . . . . . . . . . . . . . 11

22.5 Raising a power to a power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

22.6 Negative and zero indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

22.7 Fractional indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

22.8 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

22.9 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 TOPIC 1. SURDS AND INDICES

Learning objectives

By the end of this topic, you should be able to

• simplify surds;

• rationalise denominators;

• multiply and divide using positive, negative and fractional indices;

• raise a power to a power and know what a power of zero means.

© HERIOT-WATT UNIVERSITY

TOPIC 1. SURDS AND INDICES 3

1.1 Simplifying surds

Simplifying surds

Go online

A surd is an irrational number, which cannot be worked out exactly. It is a square root,cube root, etc.

For example,√2,

√7, 3√5 are all surds but

√25,

√100, 3√8 are not since

√25 = 5 and

3√8 = 2.

Work through the following examples and take note of the general rule for multiplyingsurds.

√8 =

√4× 2 =

√4×√

2 = 2√2√

12 =√4× 3 =

√4×√

3 = 2√3√

75 =√25× 3 =

√25×√

3 = 5√3√

50 =√25× 3 =

√25×√

3 = 5√2

Key point

Rule for multiplication:√a×√

b =√a× b

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples

1.

Problem:

Simplify√8

Solution:

First we look for factors of 8 that are squares.

8 can be expressed as 4 × 2.

Using the rule for multiplication gives:√8 =

√(4 × 2) =

√4√2 = 2

√2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Simplify√24

Solution:




√(4 × 6) =

√4√6 = 2

√6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

Problem:

Simplify√48



Solution:




√(4 × 12)

but 12 can also be expressed as 4 × 3 giving√48 =

√(4 × 12) =

√(4 × 4 × 3) =

√4 × √

4 × √3 = 2 × 2 × √

3 = 4√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.

Problem:

Use Pythagoras to calculate x, leaving your answer as a surd.

Solution:

x =√42 − 22 =

√16− 4 =

√12 =

√4×√

3 = 2√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplifying surds: Rule for multiplication practice

Go online

Q1: Simplify√45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q2: Simplify√72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q3: Simplify√32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q4: Simplify√50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q5: Simplify√54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q6: Simplify√80

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Key point

Rule for Division

Examples

1.

Problem:

Simplify√

205

Solution:√205 =

√4 = 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Simplify√

1875

Solution:√1875 =

√18√75

=√9×2√25×3

= 3√2

5√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

Problem:

Simplify√

1275

Solution:√1275 =

√12√75

=√4×√

3√25×√

3= 2

√3

5√3= 2

5

Note that the√3 on the numerator and denominator cancel each other out.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Simplifying surds: Rule for division practice

Go onlineQ7: Simplify

√805

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q8: Simplify√98√28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q11: Simplify√

4512

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q12: Simplify√

5430

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simplifying surds exercise

Go online

These questions are for practice only.

Q13: Simplify√27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q14: Simplify√80

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q15: Simplify√300

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q16: Simplify√44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q17: Simplify√98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q18: Simplify√125

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q19: Simplify√

169

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q20: Simplify√

2449

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q21: Simplify√

81125

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Q22: Simplify√

3218

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q23: Simplify√

7572

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Collecting like terms

Surds can also be simplified using the rules you already know from Algebra.

Key point

You know that 3x+ 5x = 8x so it follows that 3√2 + 5

√2 = 8

√2

The same rules apply to subtraction for example, 5√3− 2

√3 = 3

√3

The rules can be combined for example, 2√5 + 5

√5−√

5 = 6√5

You CANNOT simplify 3√2 + 5

√3

Collecting like terms practice: Addition and subtraction

Go online

Q24: 3√3 + 2

√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q25:√2 + 3

√2 + 7

√2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q26: 6√3− 2

√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q27: 3√5− 2

√5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q28: 5√7− 2

√7 +

√7.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Key point

We can also use the rules of simplifying surds then collecting like terms.For example,√

50 + 3√2 =

√25 ×

√2 + 3

√2

= 5√2 + 3

√2

= 8√2

Combining simplifying surds and collecting like terms practice

Go online

Q29: Simplify√18 +

√2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Q30: Simplify 7√3−√

27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q31: Simplify√32− 3

√2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q32: Simplify 2√5 +

√45−√

5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Collecting like terms exercise

Go online


Q33: Simplify√3 + 2

√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q34: Simplify 5√6− 2

√6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


√5− 2

√5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q36: Simplify 5√7−√

28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


√2−√

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Rationalising denominators

The purpose of rationalising a denominator is to turn the denominator into a wholenumber. Those of you with a modern scientific calculator will find that if you enter 1√

2

your calculator will automatically rationalise the denominator giving√22 .

The way to rationalise a surd on the denominator is to multiply both the numerator anddenominator by the surd.

Examples

1.

Problem:

Rationalise the denominator 1√2

Solution:1√2= 1×√

2√2×√

2=

√2√4=

√22

Note we multiplied by√2√2. Which simplifies to

√2√2= 1.



So multiplying by√2√2

will not change the value of 1√2

just its appearance.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Rationalise the denominator and simplify 5√10

Solution:5√10

= 5×√10√

10×√10

= 5√10√

100= 5

√10

10 =√102

Note we multiplied by√10√10

.

Remember you can simplify 510 to 1

2 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

Problem:

Rationalise the denominator and simplify 2√3−1

2√6

Solution:2√3−1

2√6

= (2√3−1)×√

6

2√6×√

6= 2

√18−√

62√36

= 2×√9×√

2−√6

2×6 = 6√2−√

612

Note we multiplied by√6√6.

Remember to multiply out the brackets.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.

Problem:

Rationalise the denominator and simplify 12+

√3

Solution:

This is the most difficult type of expression to rationalise and simplify because it has asum on the denominator.

We need to use a difference of two squares to deal with this type of expression.

Remember (x+ y)(x− y) = x2 − y2

So if we multiply the numerator and denominator by (2 − √3) we will rationalise the

denominator.

Giving 12+

√3= 1×(2−√

3)

(2+√3)(2−√

3)= 2−√

3

22−√32 = 2−√

34−3 = 2−√

31 or 2−√

3

Remember√32=

√3×√

3 =√9 = 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rationalise the denominator practice

Go online

Rationalise the denominator and simplify:

Q38: 1√6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Q39: 32√3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q40:√2+1√10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q41: 1√5−1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rationalising denominators exercise

Go online


Q42: Rationalise the denominator 1√7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q43: Rationalise the denominator and simplify 1√13

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q47: Rationalise the denominator and simplify 1+√5√

5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q48: Rationalise the denominator and simplify 1√3+1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



1.4 Multiplication and division of terms with positive indices

Using the laws of indices

Go online

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples



1.

Problem:

Simplify x3 × x5 ÷ x4

Solution:

x3 × x5 ÷ x4 = x3+5−4 = x4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Simplify x5×x4

x2 , x �= 0

Solution:x5×x4

x2 = x5+4−2 = x7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

Problem:

Simplify 3x13 × 4x

23

Solution:

3x13 × 4x

23 = 3 × 4 × x

13 × x

23

= 3 × 4 × x13+ 2

3

= 12 × x33

= 12x1

= 12x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiplication and division of indices practice

Go online

Q49: Simplify x6 × x4 ÷ x5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q50: Simplify x6 × x2

x3 , x �= 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q51: Simplify 2x14 × 5x

14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiplication and division of indices exercise

Go online



x6 , x �= 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


x4 , x �= 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Q54: Simplify x6 × 2x5

x3 , x �= 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q55: Simplify 2a12 × 4a

32

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 Raising a power to a power

How to raise a power to a power

Go online

(ax)y if x = 1 and y = 1

= (a1)1

= (a)1

= a

(ax)y if x = 4 and y = 2

= (a4)2

= (a× a× a× a)2

= (a× a× a× a)(a× a× a× a)

= a8

= a4×2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Key point

We now have three laws of indices.

1. am × an = am+n

2. am

an = am−n

3. (am)n = am×n

Example

Problem:

Simplify (a3)5.

Solution:

(a3)5 = a3×5 = a15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



Examples

1.

Problem:

Simplify (2y4)2.

Solution:

(2y4)2 = 22 × y4×2 = 4y8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Simplify (2g12 )6.

Solution:

(2g12 )6 = 26 × g

12× 6 = 64g

62 = 64g3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Raising powers practice

Go online

Q56: Simplify (a2)7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q57: Simplify (3y4)3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q58: Simplify (5m32 )2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Using three laws of indices exercise

Go online

Q59: Simplify (a6)2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q60: Simplify (2b3)2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q61: Simplify (f 4)2 × f3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q62: Simplify (4m13 )3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q63: Simplify (2n12 )2 × (3n5)2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



1.6 Negative and zero indices

Negative and zero indices

Go online

If x =0

a0 =1

If x =− 2

a−2

=1

a× a

If x =− 5

a−5

=1

a× a× a× a× a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Key point

We now have five laws of indices.

1. am × an = am+n

2. am

an = am−n

3. (am)n = am×n

4. a0 = 1

5. a−m = 1am

Examples

1.

Problem:

Simplify, giving your answer with a positive index a−5 × a4 × a−3

Solution:

a(−5)+4+(−3) = a−4 = 1a4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Problem:

Simplify, giving your answer with a positive index y7

y10

Solution:y7

y10= y7−10 = y−3 = 1

y3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

Problem:

Simplify (2g−2)3



Solution:

(2g−2)3 = 23 × g−2×3 = 8g−6 = 8g6

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Negative and zero indices practice

Q64: Simplify, giving your answer with a positive index a−3 × a5

a2

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Q65: Simplify, giving your answer with a positive index (3m−4)2

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Q66: Simplify, giving your answer with a positive index y−10 × y32

y12

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Using five laws of indices exercise

Go online

These exercises are for further practice of five laws of indices.

Q67: Simplify a−2 × a2

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Q68: Simplify (t−5)2

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Q69: Simplify (5b−7)2

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Q70: Simplify, giving your answer with a positive index f3 × f−8

f2

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Q71: Simplify, giving your answer with a positive index 2k5 × 3k−7

6k4

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Q72: Simplify, giving your answer with a positive index (2n12 )4 × (10n− 3

2 )2

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1.7 Fractional indices

The purpose of a fractional index is to define a surd.

a12 =

√a a

13 = 3

√a a

14 = 4

√a a

32 =

√a3 a

23 =

3√a2

In essence the numerator of the index is the power and the denominator is the root.

Changing fractional indices

Go online

axy = y

√ax



If x = 1 and y = 2 then a12 =

√a


√a3


3√a4


4√a4 = a

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Key point

We now have six laws of indices.

1. am × an = am+n

2. am

an = am−n

3. (am)n = am×n

4. a0 = 1

5. a−m = 1am

6. amn = n

√am

Examples

1.

Problem:

Evaluate 2512

Solution:

2512 =

√25 = 5

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2.

Problem:

Evaluate 2532

Solution:

2532 =

2√253 = 53 = 125

It does not matter whether you square root or cube the term first and since you probablydon’t know 253 it is easier to find

√25 then cube the answer.

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3.

Problem:

Evaluate 8−13

Solution:

8−13 = 1

813= 1

3√8= 1

2

• Remember a negative index moves the term onto the denominator.



• The power a third means the cube root.• The cube root of 8 is 2 because 23 = 8

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Fractional indices practice

Go online

Q73: Evaluate 912

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Q74: Evaluate 4912

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Q75: Evaluate 2713

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Q76: Evaluate 1614

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Q77: Evaluate 2723

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Q78: Evaluate 432

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Using six laws of indices exercise

Go online

These exercises are for further practice of the six laws of indices.

Q79: Evaluate 8112

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Q80: Evaluate 6413

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Q81: Evaluate 3215

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Q82: Evaluate 10032

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Q83: Evaluate 100023

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Q84: Evaluate 8134

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Q85: Evaluate 36−12

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Q86: Evaluate 16−34

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1.8 Learning points

To simplify surds.

• √63 =

√9 × √

7 = 3√7

•√

3227 =

√16×√

2√9×√

3= 4

√2

3√3

• 6√2 + 5

√2 − 3

√2 = 8

√2

To rationalise a denominator.

• 2√14

= 2×√14√

14×√14

= 2√14

14 =√147

Six laws of indices.

1. am × an = am+n

2. am

an = am−n

3. (am)n = am×n

4. a0 = 1

5. a−m = 1am

6. amn = n

√am



1.9 End of topic test

End of topic 22 test

Go online

Q87:

i. Simplify√27

ii. Simplify√80

iii. Simplify√

164

iv. Simplify√

44100

v. Simplify√

28128

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q88:

i. Simplify√2 + 3

√2

ii. Simplify 6√5 − 2

√5

iii. Simplify 3√7 − 5

√7 + 4

√7

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Q89:

i. Rationalise the denominator 3√7

ii. Rationalise the denominator 2√10

iii. Rationalise the denominator 20√20

iv. Rationalise the denominator 1√6+1

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Q90:

i. Simplify (x5 × x6)x2 , x �= 0.

ii. Simplify 2a−13 × 6a

43 , x �= 0

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Q91:

i. Simplify (2g4)3

ii. Simplify (y3)5

y2

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Q92:

i. Simplify a7 × (a−2)2

a3

ii. Simplify, giving your answer as a positive index (n12 )−2 × (n− 3

2 )2



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Q93:

i. Evaluate 12112

ii. Evaluate 1632

iii. Evaluate 8−13

iv. Evaluate 125−23

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22 GLOSSARY

Glossary

squares

Remember the squares or square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81,100,....


ANSWERS: TOPIC 22 23

Answers to questions and activities

22 Surds and indices

Simplifying surds: Rule for multiplication practice (page 4)

Q1:√45 =

√9×√

5 = 3√5

Q2:√72 =

√36×√

2 = 6√2

Q3:√32 =

√16×√

2 = 4√2

Q4:√50 =

√25×√

2 = 5√2

Q5:√54 =

√9×√

6 = 3√6

Q6:√80 =

√4×√

20 =√4×√

4×√5 = 2× 2×√

5 = 4√5

Simplifying surds: Rule for division practice (page 6)

Q7:√

805 =

√16 = 4

Q8:√98√28

=√49×√

2√4×√

7= 7

√2

2√7

Q9:√32√20

=√16×√

2√4×√

5= 4

√2

2√5= 2

√2√5

Q10:√200√50

=√100×√

2√25×√

2= 10

√2

5√2= 10

5 = 2

Q11:√

4512 =

√9×√

5√4×√

3= 3

√5

2√3

Q12:√

5430 =

√9×√

6√5×√

6=

√9√5= 3√

5

Simplifying surds exercise (page 6)

Q13: 3√3

Q14: 4√5

Q15: 10√3

Q16: 2√11

Q17: 7√2

Q18: 5√5

Q19: 43

Q20: 2√6

7

Q21: 95√5


24 ANSWERS: TOPIC 22

Q22: 43

Q23: 5√3

6√2

Collecting like terms practice: Addition and subtraction (page 7)

Q24: 5√3

Q25: 11√2

Q26: 4√3

Q27:√5

Q28: 4√7

Combining simplifying surds and collecting like terms practice (page 7)

Q29:√9×√

2 +√2 = 3

√2 +

√2 = 4

√2

Q30: 7√3−√

9×√3 = 7

√3− 3

√3 = 4

√3

Q31:√16×√

2− 3√2 = 4

√2− 3

√2 =

√2

Q32: 2√5 +

√9×√

5−√5 = 2

√5 + 3

√5−√

5 = 4√5

Collecting like terms exercise (page 8)

Q33: 3√3

Q34: 3√6

Q35: 6√5

Q36: 3√7

Q37: 8√2

Rationalise the denominator practice (page 9)

Q38: 1√6= 1×√

6√6×√

6=

√6√36

=√66

Q39: 32√3= 3×√

32√3×√

3= 3

√3

2√9= 3

√3

6 =√32

Q40:√2+1√10

= (√2+1)×√

10√10×√

10=

√20+

√10√

100=

√4×√

5+√10

10 = 2√5+

√10

10

Q41: 1√5−1

= 1×(√5+1)

(√5−1)(

√5+1)

=√5+1√

52−12

=√5+15−1 =

√5+14



Rationalising denominators exercise (page 10)

Q42:√77

Q43:√1313

Q44: 2√2

Q45:√147

Q46:√104

Q47:√5+55

Q48:√3 − 12

Multiplication and division of indices practice (page 12)

Q49: x6 × x4 ÷ x5 = x6+4−5 = x5

Q50: x6 × x2

x3 = x6 + 2 − 3 = x5

Q51:

2x14 × 5x

14 = 2 × 5 × x

14 × x

14

= 2 × 5 × x14+ 1

4

= 10 × x24

= 10x12

Multiplication and division of indices exercise (page 12)

Q52: x1 = x

Q53: x1 = x

Q54: 2x8

Q55: 8a2

Raising powers practice (page 14)

Q56: (a2)7 = a2 × 7 = a14

Q57: (3y4)3 = 33 × y4 × 3 = 27y12

Q58: (5m32 )2 = 52 × m

32× 2 = 25m

62 = 25m3



Using three laws of indices exercise (page 14)

Q59: a12

Q60: 4b6

Q61: f 11

Q62: 64m

Q63: 36n11

Negative and zero indices practice (page 16)

Q64: a(−3) + 5 − 2 = a0 = 1

Q65: (3m−4)2 = 32 ×m(−4)×2 = 9m−8 = 9m8

Q66: y(−10)+ 32− 1

2 = y(−10)+1 = y−9 = 1y9

Using five laws of indices exercise (page 16)

Q67: 1

Q68: t−10

Q69: 25b−14

Q70: 1f7

Q71: 1k6

Q72: 16n2 × 100n−3 = 1600n

Fractional indices practice (page 18)

Q73: 912 =

√9 = 3

Q74: 4912 =

√49 = 7

Q75: 2713 = 3

√27 = 3 because 33 = 27

Q76: 1614 = 4

√16 = 2 because 24 = 16

Q77: 2723 =

3√272 = 9 because 3

√27 = 3 and 32 = 9

Q78: 432 =

√43 = 8 because

√4 = 2 and 23 = 8



Using six laws of indices exercise (page 18)

Q79: 9

Q80: 4

Q81: 2

Q82: 1000

Q83: 100

Q84: 27

Q85: 16

Q86: 18

End of topic 22 test (page 20)

Q87:

i. 3√3

ii. 4√5

iii. 2

iv.√115

v.√7

4√2

Q88:

i. 4√2

ii. 4√5

iii. 2√7

Q89:

i. 3√7

7

ii.√105

iii. 2√5

iv.√6−15

Q90:

i. x9

ii. 12a

Q91:

i. 8g12

ii. y13



Q92:

i. 1

ii. 1n4

Q93:

i. 11

ii. 64

iii. 12

iv. 125


SCHOLAR Study Guide National 5 Mathematics …...TOPIC 1. SURDS AND INDICES 3 1.1 Simplifying surds...

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Transcript of SCHOLAR Study Guide National 5 Mathematics …...TOPIC 1. SURDS AND INDICES 3 1.1 Simplifying surds...