SCHOLAR Study Guide National 5 Mathematics …...TOPIC 1. SURDS AND INDICES 3 1.1 Simplifying surds...
Transcript of 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:
First we look for factors of 24 that are squares.
24 can be expressed as 4 × 6.
Using the rule for multiplication gives:√24 =
√(4 × 6) =
√4√6 = 2
√6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.
Problem:
Simplify√48
© HERIOT-WATT UNIVERSITY
4 TOPIC 1. SURDS AND INDICES
Solution:
First we look for factors of 48 that are squares.
48 can be expressed as 4 × 12.
Using the rule for multiplication gives:√48 =
√(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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 5
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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
6 TOPIC 1. SURDS AND INDICES
Simplifying surds: Rule for division practice
Go onlineQ7: Simplify
√805
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q8: Simplify√98√28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q9: Simplify√32√20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q10: Simplify√200√50
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 7
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
8 TOPIC 1. SURDS AND INDICES
Q30: Simplify 7√3−√
27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q31: Simplify√32− 3
√2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q32: Simplify 2√5 +
√45−√
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Collecting like terms exercise
Go online
These questions are for practice only.
Q33: Simplify√3 + 2
√3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q34: Simplify 5√6− 2
√6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q35: Simplify√5 + 7
√5− 2
√5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q36: Simplify 5√7−√
28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q37: Simplify√98 + 2
√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.
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 9
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
10 TOPIC 1. SURDS AND INDICES
Q39: 32√3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q40:√2+1√10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q41: 1√5−1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rationalising denominators exercise
Go online
These questions are for practice only.
Q42: Rationalise the denominator 1√7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q43: Rationalise the denominator and simplify 1√13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q44: Rationalise the denominator and simplify 4√2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q45: Rationalise the denominator and simplify 2√14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q46: Rationalise the denominator and simplify 52√10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q47: Rationalise the denominator and simplify 1+√5√
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q48: Rationalise the denominator and simplify 1√3+1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 11
1.4 Multiplication and division of terms with positive indices
Using the laws of indices
Go online
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples
© HERIOT-WATT UNIVERSITY
12 TOPIC 1. SURDS AND INDICES
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
These questions are for practice only.
Q52: Simplify x4 × x3
x6 , x �= 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q53: Simplify x3 × x2
x4 , x �= 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 13
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
14 TOPIC 1. SURDS AND INDICES
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 15
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
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3.
Problem:
Simplify (2g−2)3
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16 TOPIC 1. SURDS AND INDICES
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q65: Simplify, giving your answer with a positive index (3m−4)2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q66: Simplify, giving your answer with a positive index y−10 × y32
y12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using five laws of indices exercise
Go online
These exercises are for further practice of five laws of indices.
Q67: Simplify a−2 × a2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q68: Simplify (t−5)2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q69: Simplify (5b−7)2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q70: Simplify, giving your answer with a positive index f3 × f−8
f2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q71: Simplify, giving your answer with a positive index 2k5 × 3k−7
6k4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
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TOPIC 1. SURDS AND INDICES 17
If x = 1 and y = 2 then a12 =
√a
If x = 3 and y = 2 then a32 =
√a3
If x = 4 and y = 3 then a43 =
3√a4
If x = 4 and y = 4 then a44 =
4√a4 = a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.
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18 TOPIC 1. SURDS AND INDICES
• The power a third means the cube root.• The cube root of 8 is 2 because 23 = 8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fractional indices practice
Go online
Q73: Evaluate 912
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q74: Evaluate 4912
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q75: Evaluate 2713
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q76: Evaluate 1614
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q77: Evaluate 2723
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q78: Evaluate 432
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using six laws of indices exercise
Go online
These exercises are for further practice of the six laws of indices.
Q79: Evaluate 8112
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q80: Evaluate 6413
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q81: Evaluate 3215
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q82: Evaluate 10032
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q83: Evaluate 100023
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q84: Evaluate 8134
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Q85: Evaluate 36−12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q86: Evaluate 16−34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 19
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
© HERIOT-WATT UNIVERSITY
20 TOPIC 1. SURDS AND INDICES
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q90:
i. Simplify (x5 × x6)x2 , x �= 0.
ii. Simplify 2a−13 × 6a
43 , x �= 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q91:
i. Simplify (2g4)3
ii. Simplify (y3)5
y2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Q92:
i. Simplify a7 × (a−2)2
a3
ii. Simplify, giving your answer as a positive index (n12 )−2 × (n− 3
2 )2
© HERIOT-WATT UNIVERSITY
TOPIC 1. SURDS AND INDICES 21
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Q93:
i. Evaluate 12112
ii. Evaluate 1632
iii. Evaluate 8−13
iv. Evaluate 125−23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
© HERIOT-WATT UNIVERSITY
22 GLOSSARY
Glossary
squares
Remember the squares or square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81,100,....
© HERIOT-WATT UNIVERSITY
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
© HERIOT-WATT UNIVERSITY
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
© HERIOT-WATT UNIVERSITY
ANSWERS: TOPIC 22 25
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
© HERIOT-WATT UNIVERSITY
26 ANSWERS: TOPIC 22
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
© HERIOT-WATT UNIVERSITY
ANSWERS: TOPIC 22 27
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
© HERIOT-WATT UNIVERSITY
28 ANSWERS: TOPIC 22
Q92:
i. 1
ii. 1n4
Q93:
i. 11
ii. 64
iii. 12
iv. 125
© HERIOT-WATT UNIVERSITY