chapter 07 new century maths

Number/Patterns and algebra

Indices

7

The speed of light is about 300 000 km/s. In one year, light travels approximately 9 460 000 000 000 km. The light from the stars travels for many years before it is seen on Earth. Powers or indices provide a way to work easily with very large numbers or with very small numbers.

07_NC_Maths_9_Stages_5.2/5.3 Page 218 Friday, February 6, 2004 2:17 PM

■ describe and evaluate numbers written in index form, using terms such as ‘base’, ‘power’, ‘index’ and ‘exponent’

■ develop and use the index laws for multiplying and dividing terms with the same base, and for the power of a base raised to a power

■ develop and use zero and negative indices■ use fractional indices for square roots and cube roots■ express and order numbers in scientific notation■ convert numbers expressed in scientific notation to decimal form■ enter and read scientific notation on a calculator■ calculate with numbers expressed in scientific notation.

■ base A number that is raised to a power, meaning that it is multiplied by itself repeatedly. For example, in 25, the base is 2.

■ power The number of times a base is multiplied by itself. For example, 25 means 2 × 2 × 2 × 2 × 2, and is 2 to the power of 5. A power is also called an index or an exponent.

■ index notation or index form Repeated multiplication written in the form an. For example, 2 × 2 × 2 × 2 × 2 written using index notation is 25.

■ negative power A power that is a negative number, as in the expression 3−2.

■ scientific notation A shorter way of writing very large or very small numbers using powers of 10. For example, 9 460 000 000 000 in scientific notation is 9.46 × 1012.

The story is that Sissa ben Dahir, who invented chess, was offered any reward he wanted by the Indian King Shirham. Sissa asked for the following:

‘I will have one grain of wheat for the first square of my chessboard, two grains of wheat for the second, four for the third and so on to the sixty-fourth square.’

King Shirham granted his request without thinking!■ How many grains of wheat would be needed for the 64th square?■ How many grains of wheat would be needed altogether to meet Sissa’s

request?■ If a grain of wheat weighs 100 mg, how many tonnes of wheat would there

be on the chessboard?

In this chapter you will:

Wordbank

Think!

I ND I C ES 219

CHAPTER 7

07_NC_Maths_9_Stages_5.2/5.3 19 Friday, February 6, 2004 2:17 PM

220

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

Powers

The numbers 2, 4, 8, 16, … are powers of 2. (They can also be written as 2

1

, 2

2

, 2

3

, 2

4

,… .)

Similarly, the numbers 3, 9, 27, 81, 243, … are powers of 3.

1 Evaluate:a 4 × 4 b 2 × 2 × 2 × 2 × 2 c 3 × 3d 5 × 5 × 5 e 10 × 10 × 10 × 10 × 10 f 7 × 7g 4 × 4 × 4 h 8 × 8 × 8 × 8 × 8 × 8 i 12 × 12 × 12 × 12

2 Evaluate:a 43 b 102 c 26 d 34 e 42 f 106

3 Express in index form:a 5 × 5 b 4 × 4 × 4 × 4 × 4 c 6 × 6 × 6d 3 × 3 × 3 × 3 × 3 × 3 e y × y f m × m × m × m × mg a × a × a h x × x × x × x × x × x i d × d × d × d

4 Write in expanded form:a 103 b 82 c 15 d 24 e 31

f k2 g w4 h d5 i p1 j c3

5 Evaluate:

a b c d e

6 Evaluate:

a b c d e f

400 289 1024 225 625

83 273 -83 -2163 10003 -273

Start up

Worksheet 7-01

Brainstarters 7

Working mathematicallyReasoning and reflecting: Powers and the power keyNumbers expressed as powers of numbers, such as 27, can be easily evaluated using the

power key ( or or ) on your calculator.

1 a Evaluate 24 = 2 × 2 × 2 × 2 = ?b Evaluate 24 using the power key on your calculator as follows:

2 4

(Note: Your answers for parts a and b should be the same.)

2 Use the power key to evaluate each of the following. Compare your answers to those of other students.a 45 b 77 c 34 d 118

3 a Copy the table below into your book and use your calculator to evaluate the first six powers of 4, 5, 6 and 7, and enter them in your table. Compare your results with those of other students in your class.

^ xy yx

^ =


I ND I C ES 221 CHAPTER 7

Index notationConsider 2 × 2 × 2 × 2 × 2 = 25.

2 × 2 × 2 × 2 × 2 is the expanded or factor form .

25 is the index notation or exponent form.

In 25, 5 is called the index or the power or the exponent.

The base is 2.

25 is read as ‘2 to the power of 5’ or ‘2 to the 5th’.

b Using the results in your table, evaluate:i 81 ii 151 iii 21 iv 231

c What is the value of a1 (that is, any number to the power of 1)?

4 a Evaluate the powers of 2 (21, 22, 23, …). What is the largest power of 2 that your calculator can display as a whole number?

b Find the largest power of each of the following numbers that your calculator can display as a whole number.

i 3 ii 4 iii 5 iv 6 v 7Compare your results with those of other students in your class.

Powers of 4 Powers of 5 Powers of 6 Powers of 7

41 = 51 = 61 = 71 =

42 = 52 = 62 = 72 =

� � � �

46 = 56 = 66 = 76 =

Skillsheet 7-01Indices

25 index, power or exponent

base

SkillBuilder 11-01

Introduction to indices

Example 1

Express in index form:a 3 × 3 × 3 × 3 b m × m × m × m × m c a × a × … × a

Solutiona 3 × 3 × 3 × 3 = 34 b m × m × m × m × m = m5 c a × a × … × a = an

Express in index form:a 5 × 5 × 5 × 6 × 6 × 6 × 6 b p × p × p × p × t × t × t × t × t × tc a × a × … × a × b × b × … × b

Solutiona 5 × 5 × 5 × 6 × 6 × 6 × 6 = 53 × 64

b p × p × p × p × t × t × t × t × t × t = p4 × t6 = p4t6

n factors

4 factors

5 factors

n factors

Example 2

n factors

m factors

3 factors

4 factors

4 factors 6 factors


222 NEW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

c a × a × … × a × b × b × … × b = an × bm = anbm

Express in expanded form:a 35 b 5x3m4

Solutiona 35 = 3 × 3 × 3 × 3 × 3 b 5x3m4 = 5 × x × x × x × m × m × m × m

n factors

m factors

Example 3

1 For each of the following:i state the base ii state the index iii write the expression in words.a 37 b 73 c k4 d 4k e an

2 Express in index form:a 5 × 5 × 5 × 5 b 10 × 10 c 8 × 8 × 8 d 32 × 32e 9 × 9 × 9 × 9 × 9 f 12 × 12 × 12 g 1 × 1 × 1 × 1 h 6

3 Write using index notation:a a × a × a × a b m × m c y × y × y × yd q × q × q × q × q × q e p × p × p f w

4 Write these in index notation:a 3 × 3 × 2 × 2 × 2 × 2 × 2 b 3 × 3 × 3 × 3 × 7 × 7 × 7 c 5 × 5 × 5 × 5 × 5 × 5 × 8 × 8d 6 × 6 × 6 × k × k e x × y × x × y × x f 5 × n × 5 × n × n

5 Write in expanded or factor form:a 64 b 103 c 64 × 103 d p4

e 5p4 f 52p4 g p4q5 h 5p4q5

i 52p4q5 j ab3 k ab3c2 l a4bc2

m m3n4 n 2y3d2 o 42a3m p w4y2v3

6 Evaluate the following.a 24 b 33 c 52 d 43

e 27 f 53 g 132 h 83

i 64 j 73 k 210 l 35

m 52 × 55 n 33 × 104 o 44 × 62 p 35 × 53

7 Evaluate, correct to three decimal places:

a 3.17 b (0.145)2 c d (−2.5)7

e (1.1)5 f g h (0.18)2

8 Find the missing powers in:a 8 = 2? b 81 = 3? c 216 = 6? d 144 = 12?

e 4096 = 2? f 2401 = 7? g 64 = 2? h 625 = 5?

9 Evaluate, correct to 2 significant figures:a 126 b (−11)5 c 212

d (3.1)3 e (−1.11)2 f (7.2)4

−25---

4

127---

4−2

3---

5

Exercise 7-01

Example 1

Example 2

Example 3

CAS 7-01

Index form



10 If p = 4, q = 3, r = 5, evaluate the following.a p4 b pq3 c (pq)3

d pq2r e (pqr)2 f

11 a Evaluate the terms of the pattern (−1)1, (−1)2, (−1)3, (−1)4, (−1)5, (−1)6, …Write down what you observe about the odd and even powers and the sign of the answer.

b Without using a calculator, find the value of:i (−1)98 ii (−1)99

c Predict the values of the following.i (−1)n when n is even ii (−1)n when n is odd

d Hence evaluate:i (−1)1 + (−1)2 + (−1)3 + (−1)4 + … + (−1)36

ii (−1)1 + (−1)2 + (−1)3 + (−1)4 + … + (−1)37

12 a Evaluate the following.i 62 ii 662 iii 6662 iv 66662

b Predict the values of the following.i (666 666)2 ii (666 666 666)2

qr---

3

Working mathematicallyApplying strategies and reasoning: Cell growthUse a spreadsheet to help you with this investigation.

Over the centuries, millions of people have contracted diseases such as smallpox, typhoid and diphtheria. These diseases start off as a few cells that multiply at an alarming rate until there are too many in the body, causing the person to become ill. In some cases this can be fatal.

Suppose one of these diseases grows by the cells splitting into equal parts every 10 seconds; that is, every 10 seconds, the number of cells doubles.

Disease A

1 Starting with one cell, calculate the cell population after:a 30 s b 40 s c 1 min d 1 min 30 se 2 min f 3 min g 4 min h 4 min 20 s

t = 0 s 1 cell

2 cells

4 cells

t = 10 s

t = 20 s

Spreadsheet 7-01

Cell growth



The index laws

2 Starting with one cell, find how long it will take until there are:a 64 cells b 256 cellsc over 500 cells d over 1000 cellse over 3000 cells f over 10 000 cellsg over 1 000 000 cells h over 40 000 000 cells

Further diseases are discovered that multiply at different rates:• Disease B: A single cell divides into three identical cells every 15 seconds.• Disease C: A single cell divides into four identical cells every 20 seconds.• Disease D: A single cell divides into five identical cells every 30 seconds.

3 Expand your spreadsheet to show all four disease strains, and answer Questions 1 and 2 for strains B, C and D.

4 If the diseases all begin from one cell at time 0 seconds, when does the growth of each strain pass that of the others?

5 Graph the growth of each disease on your graphics calculator or by using the Graph option in the spreadsheet.

6 In your own words, describe in writing the shape of the exponential graph generated for each disease.

Working mathematicallyReasoning and reflecting: Multiplying terms with the same base1 Use a calculator to find the value of:

a i 23 × 24 ii 27 b i 35 × 33 ii 38

c i 44 × 42 ii 46 d i 56 × 53 ii 59

2 What do you notice about each pair of answers in Question 1?

3 Is it true that 25 × 27 = 212? Explain.

4 State whether each of the following are true (T) or false (F).Explain each choice.a 26 × 24 = 210 b 74 × 78 = 732 c 45 × 48 = 440 d 37 × 312 = 319

5 Copy and complete the following.a 47 × 43 = 4… b 53 × 54 = 5… c 68 × 65 = 6…

d 83 × 82 = … e k3 × k8 = k… f m3 × m7 = …

6 Use a calculator to find the value of:a i 23 × 25 ii 48 b i 54 × 56 ii 2510

c i 37 × 34 ii 911 d i 62 × 63 ii 365

7 Use your results from Question 6 to decide whether these are true (T) or false (F):a 23 × 25 = 48 b 54 × 56 = 2510 c 37 × 34 = 911 d 62 × 63 = 365

8 Write true (T) or false (F) for each of the following.a 53 × 58 = 2511 b 27 × 210 = 217 c 73 × 72 = 75

d 43 × 410 = 430 e 53 × 54 = 257 f 33 × 39 = 312



Law 1: Multiplying terms with the same baseConsider 24 × 23 = (2 × 2 × 2 × 2) × (2 × 2 × 2)

= 2 × 2 × 2 × 2 × 2 × 2 × 2= 27

But 24 + 3 = 27

∴ 24 × 23 = 24 + 3 = 27

Proof:am × an =

=

= am + n

SkillBuilder 11-02

The first index law

When multiplying terms with the same base, add the powers:

am × an = am + n

a a …× a a a … a××××××

n factorsm factors

a a …× a××m n factors+

Example 4

Simplify the following, expressing your answers in index form.a 63 × 67 b 5 × 53 c y4 × y8

Solution

Simplify the following.a 3p4 × 2p6 b 5e2f × 3e4f 5

Solution

a 63 × 67 = 63 + 7

= 610

b 5 × 53 = 51 × 53

= 51 + 3

= 54

c y4 × y8 = y4 + 8

= y12

a 3p4 × 2p6 = (3 × 2) × (p4 × p6)= 6p4 + 6

= 6p10

b 5e2f × 3e4f 5 = (5 × 3) × (e2 × e4) × (f × f 5)= 15e2 + 4f 1 + 5

= 15e6f 6

Example 5

1 Simplify (giving answers in index notation):a 103 × 102 b 10 × 104 c 32 × 35 d 74 × 7e 8 × 83 × 84 f 54 × 5 × 54 g 6 × 62 × 63 × 64 h 44 × 44 × 44

i 117 × 1113 j 2 × 23 k 34 × 3 × 37 l 72 × 75 × 7

2 Simplify:a x × x4 b g4 × g4 c w7 × w d b3 × b10

e p10 × p10 f r × r g y × y3 × y2 h m3 × m × m4

Exercise 7-02

CAS 7-02Index

multiplication

Example 4



Law 2: Dividing terms with the same base

Consider 57 ÷ 54 =

=

= 5 × 5 × 5

= 53

3 Simplify:a 3p2 × 2p5 b 4y10 × 3y2 c 6m × 3m8

d h3 × 5h8 e 3q3 × 8q8 f 2a2 × 5a5

g 5n8 × 6n8 h 2b3 × 15b6 i 3e4 × e6

j 10p3 × 5p2q k 8a3b2 × 2b3a l 4w5y2 × 5w4y3

m 5a3c2 × 2b4c n 10p3q8 × qp2 o 4g3h2 × 5gh4

4 Write true (T) or false (F) for each of the following.a 53 × 37 = 1510 b 72 × 82 = 564 c 3 × 72 = 212 d 43 × 47 = 410

e 32 × 24 × 32 × 25 = 34 × 29 f 52 × 53 = 2515 g 27 × 28 = 215

h 73 × 75 = 498 i 42 × 33 = 126 j 54 × 32 × 37 × 5 = 39 × 55

5 Simplify and evaluate:a 23 × 25 b 23 × 52 c 102 × 210 d 53 × 35

e 33 × 33 f 53 × 23 g 102 × 103 h 210 × 103

6 Simplify:a x4 × x3 × x2 b y6 × x3 × y c 5 × 3n × 4n2 d 5 × m × 4n2

e 5qp × 4q2 × 5p3 f (a4 × b3) × (a4 × b2) g 4a × 4b h 2x + 1 × 2x

i 32y × 3y j (p + q)2 × (p + q)3 k (x – y) × (x – y)2 l (a + 3)n × (a + 3)

Example 5

SkillBuilder 11-06

Multiplying terms with

indices

Working mathematicallyReasoning and reflecting: Dividing terms with the same base1 Use a calculator to find the value of:

a i 210 ÷ 27 ii 23 b i 55 ÷ 53 ii 52

c i 37 ÷ 32 ii 35 d i 68 ÷ 64 ii 64

2 What do you notice about each pair of answers?

3 Is it true that 38 ÷ 36 = 32? Explain.

4 State whether each of the following are true (T) or false (F).a 310 ÷ 36 = 34 b 48 ÷ 42 = 44 c 212 ÷ 23 = 24 d 610 ÷ 65 = 65

5 Copy and complete the following.a 27 ÷ 23 = 2… b 58 ÷ 56 = 5… c 67 ÷ 62 = 6…

d 311 ÷ 36 = … e y8 ÷ y5 = … f m12 ÷ m10 = …

6 Write true (T) or false (F) for the following.a 106 ÷ 102 = 104 b 106 ÷ 102 = 13 c 106 ÷ 102 = 103

d 812 ÷ 83 = 89 e 410 ÷ 45 = 42 f 66 ÷ 62 = 64

SkillBuilder 11-03

Division of terms with indices

57

54-----

5 5 5 5 5 5 5××××××5 5 5 5×××

----------------------------------------------------------1

1 1 1

1 1 1 1



But 57 − 4 = 53

∴ 57 ÷ 54 = = 57 − 4 = 53

Proof:

am ÷ an =

=

= a × a × … × a(m − n factors)= am − n

57

54-----

When dividing terms with the same base, subtract the powers:

am ÷ an = = am − nam

an------

am

an------

a a a a a … a××××××a a a a … a×××××

-------------------------------------------------------------(m factors)(n factors)

1 1

1 1

Example 6

Simplify the following, expressing your answers in index form.

a 45 ÷ 43 b c y12 ÷ y3

Solutiona 45 ÷ 53 = 45 − 3 b = 107 − 4 c y12 ÷ y3 = y12 − 3

= 42 = 103 = y9

Simplify the following.a k7 ÷ k b 15m8 ÷ 3m2 c 30a5b7 ÷ 10a2b5

Solutiona k7 ÷ k = k7 ÷ k1

= k7 − 1

= k6

b 15m8 ÷ 3m2 =

= 5m8 − 2

= 5m6

c 30a5b7 ÷ 10a2b5 =

= 3a5 − 2b7 − 5

= 3a3b2

107

104--------

107

104--------

Example 7

15m8

3m2-------------

5

1

30a5b7

10a2b5-----------------

1

3

Just for the recordRemember that taxiIndian mathematician Srinivasa Ramanujan (1888–1920) loved working with numbers. One day he was visited by a friend in a taxi numbered 1729. When Ramanujan heard the number, he immediately said ‘1729 is a very interesting number as it is the smallest number that can be expressed as the sum of two cubes in two different ways.’

This means that we can write 1729 = x3 + y3

Here is one of the possible ways:1729 = 103 + 93

Find the other.



1 Simplify, giving your answers in index form:

a b c d 74 ÷ 73

e 105 ÷ 105 f 85 ÷ 8 g 2015 ÷ 205 h

i 680 ÷ 620 j 815 ÷ 811 k 312 ÷ 36 l 218 ÷ 211

2 Simplify:

a b c d b16 ÷ b15

e m16 ÷ m16 f n7 ÷ n g t18 ÷ t9 h

i e30 ÷ e10 j k p15 ÷ p10 l w24 ÷ w6

3 Simplify the following.a 10y15 ÷ 5y3 b 20w9 ÷ 4w3 c 24r 8 ÷ 3r 2 d

e f g 16h10 ÷ 8h h 15y8 ÷ 15y4

i 18g60 ÷ 6g4 j k l

m 20m15n ÷ 2m14 n 36y8x7 ÷ 12x3y o 44e4f 10 ÷ 4ef2 p 30k7m4 ÷ 6k6m2

4 Write true (T) or false (F) for the following.a 103 ÷ 22 = 51 b 84 ÷ 44 = 22 c 1210 ÷ 1210 = 1 d 158 ÷ 154 = 152

e 109 ÷ 103 = 106 f 74 ÷ 72 = 12 g ÷ 42 h 123 ÷ 33 = 41

5 Evaluate:a 210 ÷ 25 b 45 ÷ 23 c 33 ÷ 23 d e f

g 45 ÷ 210 h 203 ÷ 53 i 106 ÷ 54 j 49 ÷ 83 k l 310 ÷ 272

6 Simplify:

a b c

d 4a ÷ 4b e 2x + 1 ÷ 2x f 32y ÷ 3y

g h i ×

58

52----- 912

93------- 227

23-------

220

2-------

h20

h4------- y8

y2----- a12

a4-------

w25

w--------

d9

d5-----

30x4

x3-----------

10m10

2m2--------------- 12g12

6g6--------------

a6b3

a2b2----------- 36p8q3

4p4q------------------ 100f 2g4

5 f g2---------------------

204

52--------

103

23-------- 54

54----- 210

52-------

125

68--------

x4 x3×x2

---------------- y10

y3 y×-------------- a5 a3×

a a4×-----------------

4m3 5m7×10m6

-------------------------- 6n16 8n4×3n3 4n5×-------------------------- p6

6p2--------- 30p4

5p------------

Exercise 7-03Example 6

Example 7

CAS 7-03

Index division

SkillBuilder 11-04

Using the second index law

Working mathematicallyReasoning and reflecting: Powers to powers1 Use a calculator to find the value of:

a i (23)2 ii 26 b i (34)3 ii 312

c i (52)3 ii 56 d i (25)4 ii 220

2 What do you notice about each pair of answers in Question 1?



Law 3: Raising a power to a powerConsider (42)5 = 42 × 42 × 42 × 42 × 42

= (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4)= 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4= 410

But 42 × 5 = 410

∴ (42)5 = 42 × 5 = 410

Proof:

Law 4: Powers of products and quotientsConsider (2 × 3)5 = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3)

= 2 × 2 × 2 × 2 × 2 × 3 ×3 × 3 × 3 × 3

= 25 × 35

3 Is it true that (27)3 = 221? Explain.

4 State whether each of the following is true (T) or false (F).a (35)3 = 315 b (23)2 = 25 c (210)4 = 214

d (42)5 = 410 e (33)6 = 318 f (52)4 = 56

5 Copy and complete:a (37)2 = 3… b (52)6 = 5… c (45)2 = 4…

d (a3)4 = a… e (83)7 = … f (k4)6 = …

6 State whether the following are true (T) or false (F).a (25)7 = 212 b (28)3 = 224 c (53)4 = 512

d (73)7 = 77 e (84)5 = 89 f (66)5 = 630

When raising a term with a power to another power, multiply the powers:(am)n = am × n

(am)n = am × am × … × am

= a × a × … × a × a × a × … × a × … × a × a × … × a

= am × n

n factors

m factors m factors m factors

n lots of m factors

Example 8

Simplify the following, expressing your answers in index form.a (23)5 b (y2)14

Solutiona (23)5 = 23 × 5

= 215b (y2)14 = y2 × 14

= y28

SkillBuilder 11-07

Multiplying expressions with

brackets



Also =

=

=

Proof:(ab)n =

= a × b × a × b × … × a × b

=

= an × bn

Also = × × … ×

=

=

53---

4 53--- 5

3--- 5

3--- 5

3---×××

5 5 5 5×××3 3 3 3×××------------------------------

54

34

-----

Powers of products and quotients:

(ab)n = anbn and ab---

n an

bn------=

ab ab …× ab××n factors

a a …× a××n factors

b b …

b ××× n factors

×

ab---

n ab--- a

b--- a

b---

n factors

a a … a×××b b b … b××××------------------------------------------- n factors( )

n factors( )

an

bn

-----

Example 9

Simplify each of the following.

a (2k)5 b (5m4)3 c d

Solutiona (2k)5 = 25 × k5

= 32k5

b (5m4)3 = 53 × (m4)3

= 125 × m4 × 3

= 125m12

c =

=

d =

=

=

m4----

3 2w3

3---------

4

m4----

3 m3

43------

m64------

3

2w3

3---------

4 2w3( )34

--------------4

24 w3( )4×34

-------------------------

16w12

81---------------



Just for the recordThe house flyThe female common house fly, Musca domestica, can lay up to 1000 eggs at a time. In three weeks these reach maturity and are ready to breed. Huge populations would result if all the descendants of a single pair of house flies survived and reproduced. Fortunately, this is not the case as the mortality rate is very high. The few house flies we see are the true survivors.

Over the 13 weeks of summer, how many descendants could a single pair of house flies produce, assuming that each pair (original and descendants) mates only once?

(Give your answer in index form.)

1 Simplify, giving your answers in index form:a (43)2 b (52)8 c (33)4 d (27)4 e (21)2

f (94)3 g (100)2 h (64)5 i (53)5 j (25)10

k (31)5 l (73)0 m (22)10 n (132)2 o (44)4

2 Simplify each of the following. Give your answers in index form.a (e2)4 b (t5)5 c (y3)7 d (c)5 e (m7)5

f (y4)4 g (h0)6 h (p6)3 i (w4)1 j (x1)10

k (n3)8 l (d3)3 m (k5)10 n (d3)4 o (a8)8

3 Simplify the following.a (2d)4 b (5m)2 c (4y5)2 d (3x2)4 e (5m6)5

f (2w5)3 g (10d5)4 h (3e7)3 i (2b4)1 j (6d6)2

k (3f 4)5 l (2c3)10 m (3h5)4 n (6k)2 o (8w3)2

4 Simplify each of the following.

a b c d

e f g h

i j k l

5 Simplify the following, giving your answers in index form.a (m3)10 b (5t)3 c (−2)8 d (−x)3

e (y3)12 f (4w5)4 g (−2d)5 h (210)10

i (−3p2)3 j (−5m3)2 k (3f 5)5 l (−m2)4

6 Evaluate:a (23)2 b (−32)2 c (102)3 d (−5)2

e (−2)3 f (−42)3 g (−34)2 h (−52)3

i j k l

e2---

5 x7---

2 3m2

------- 3 5h

6------

2

f2

3-----

4

n5

p2

----- 8 w

2

t3

------ 5 am

c-------

4

2k3

5--------

2

3r4

c2

-------- 2 a

2b

d5

-------- 4 5c

2

3x3

-------- 3

32---

2 25---

2 52---

3−3

4---

2


Example 9

Worksheet7-02

Indices puzzle

SkillBuilder 11-08

The fourth index law




a (l3m5)6 b (x2y4)5 c d

e (−we2k3)3 f g h (w2ak4)8

i (−3m2n)5 j (−2p2w3)4 k (a2d3y5)0 l

m n o (4k3m)3 p (8k4y5)2

q (3a3df4)5 r (6d5p2)4 s t

8 Simplify:a (52)x b (5x)2 c ((x + 1)2)3

d (b3)4 ÷ (b4)2 e (b3)4 × (b4)2 f (6n4)2 × (3n2)2

g (6n4)2 ÷ (3n2)2 h i

a4

d3

----- 7

−m2

n------

4

2y3

x2

-------- 3

p2q

3

t4

----------- 5

a2c

4

d7

----------- 0

d2e

5f

4---------------

3

−m2n

3

2y5

------------ 5

−3ay4

b2

------------ 5 k p

3

3q4

--------- 2

3x5( )3

x2( )4

---------------- 5y5

2y( )5×8y

2y

3( )2×----------------------------

Working mathematicallyQuestioning and reasoning: The power of zero1 Copy and complete the following sentence.

A number remains unchanged when multiplied or divided by …

2 Copy and complete the following.a 34 × 30 = 3? b 52 × 50 = 5? c 20 × 27 = 2? d 70 × 73 = 7?

e 45 × 40 = 4? f 50 × 57 = 5? g 30 × 35 = ? h 80 × 86 = ?

3 Copy and complete the following.a 25 ÷ 20 = 2? b 35 ÷ 30 = 3? c 42 ÷ 40 = 4? d 93 ÷ 90 = 9?

e 56 ÷ 50 = 5? f 84 ÷ 80 = 8? g 157 ÷ 150 = ? h 68 ÷ 60 = ?

4 Copy and complete the following tables. Compare your answers with those of other students.

5 Look at your results from Questions 1 to 4. Can you suggest a value for any number (or base) raised to the power of zero (for example, 30 = ?, 50 = ?)? Explain.

a Indexform

Numberb Index

formNumber

c Indexform

Number

25 32 35 243 85

24 16 34 84

23 33 83

22 32 82

21 31 81

20 30 80



The zero indexConsider 53 ÷ 53 =

=

= 1But 53 ÷ 53 = 53 − 3 = 50

∴ 50 = 1

Proof:

am ÷ am =

= 1

but am ÷ am = am − m = a0

∴ a0 = 1

SkillBuilder 11-05

Raising to the power of 0

53

53

-----

5 5 5××5 5 5××---------------------

1 1 1

1 1 1

Any number raised to the power of zero is equal to 1:

a0 = 1

a a a … a××××a a a … a××××-------------------------------------------

1

1 1 1 1

1 1 1

(m factors)(m factors)

Example 10

Simplify the following.a 70 b (−3)0 c m0

Solutiona 70 = 1 b (−3)0 = 1 c m0 = 1

Simplify:a (ab)0 b (5k)0

Solutiona (ab)0 = 1 b (5k)0 = 1

Simplify:a 5d0 b (3y)0 + 3y0

Solutiona 5d0 = 5 × d0

= 5 × 1= 5

b (3y)0 + 3y0 = 1 + 3 × y0

= 1 + 3 × 1= 4

Example 11

Example 12



1 Simplify the following.a 80 b (−2)0 c d0 d m0

e f (−6)0 g (−700)0 h (1 000 000)0

i (−14)0 j k a0 l

2 Simplify:a (km)0 b (x2y)0 c (xyw)0 d (−ab)0

e f g (7y)0 h (9cd)0

3 Simplify the following.a 70 + 20 b 3y0 c −(4m)0 d 3 × (5d)0

e (5t2)0 f (6x)0 + 20 g 2m0 + (2m)0 h 2w0 × 3p0

i 12u0 ÷ 3 j 32 × 50 k (5a)0 + 4 l 8b0 − (3b)0

m 6h0 − (6h)0 n −7c + 4c0 o (3e2)0 − (10e)0 p

q 1000 − 10000 r 3f 0 + 4 − (5f)0 s 36q5 ÷ 12q5 t (3x3)3 ÷ x9

u 60m5n3 ÷ 12mn3 v 12p0 ÷ (2p)0 w (a2b3)0 x 7 × 4k0

23---

0

54---

0−12---

0

pq---

0 34---

0

12---

0 12---y0+


Example 11

Example 12

Working mathematicallyApplying strategies and reasoning: Negative powers1 a Copy and complete the following table of descending powers of 10. Use your

calculator if necessary. (Don’t be alarmed if the calculator gives decimal answers.) What rule did you use to complete the pattern?

b To see the hidden pattern clearly, you will need to change the decimals into fractions. Copy and complete the following table. Express each decimal as a fraction, then write it as a power of 10. (The first two have been done for you.)

c Look carefully at the fractions written as powers of 10. What do you notice when you compare them with the corresponding negative powers of 10? Write down your findings. Write 10−7 and 10−8 as fractions using the power of 10.

d What does this tell you about negative powers?e Write down what you have learnt about raising a number to a negative power.

Powers to ten 106 … 100 10−1 10−2 … 10−6

Decimal form 1 000 000 … 0.1 0.01 …

Powers of ten 10−1 10−2 … 10−6

Decimal form 0.1 0.01 …

Fraction form …

Fraction form with powers of ten

…

110------ 1

100---------

1

101

-------- 1

102

--------

SkillBuilder 11-09

Exercising the four index laws



The negative indexConsider 24 ÷ 27 =

=

=

=

2 a Copy and complete the table below. Use your calculator to express each power of 5 as whole numbers or a fraction.

b 5−2 can be written as , or as . Use the table to write each of the following in two ways.i 5−3 ii 5−4 iii 5−5

c Write each of the following in two ways.i 4−2 ii 7−3 iii 2−6

3 Consider: =

=

=

But = 104 ÷ 107

= 10−3 (Using Law 2)

So 10−3 =

Using this method, simplify (in the two ways):

a to show that 2−5 = b to show that 3−1 =

c to show that 5−6 = d to show that a−2 =

4 The reciprocal of 35 is = 3−5.

Use negative indices to write the reciprocals of the following.a 24 b 52 c 4 d k5 e m3

Compare your answers with those of other students.

53 52 51 50 5−1 5−2 5−3 5−4 5−5

125 125------ 1

52-----=

125------ 1

52-----

104

107-------- 10 10 10 10×××

10 10 10 10 10 10 10××××××-------------------------------------------------------------------------------

110 10 10××------------------------------

1

103--------

104

107--------

1

103--------

23

28----- 1

25----- 34

35----- 1

3---

52

58----- 1

56----- a4

a6----- 1

a2-----

1

35-----

SkillBuilder 11-13

Division with a larger index in

the denominator

24

27-----

2 2 2 2×××2 2 2 2 2 2 2××××××----------------------------------------------------------

1 1 1

1 1 1 1

1

12 2 2××---------------------

1

23-----



But 24 ÷ 27 = 24 − 7

= 2−3

∴ 2−3 =

Proof: a0 ÷ an =

= (since a0 = 1)

But a0 ÷ an = a0 − n

= a−n

∴ a−n =

1

23-----

A negative power or index gives a fraction (with numerator 1):

a−m = 1

am------

a0

an-----

1

an-----

1

an-----

Example 13

Express using positive indices:a 3−1 b 4−3 c k−5

Solutiona 3−1 = b 4−3 = c k−5 =

=

Express using positive indices:a 3k−5 b a2b−3 c (5m)−2

Solution

Evaluate 2−3, leaving your answer as a fraction.

Solution

2−3 =

=

=

a 3k −5 = 3 × k−5

= × (3 = )

=

b a2b−3 = a2 × b−3

= ×

=

c (5m)−2 =

=

1

31----- 1

43----- 1

k5-----

13---

Example 14

31--- 1

k5----- 3

1---

3

k5-----

a2

1----- 1

b3-----

a2

b3-----

1

5m( )2---------------

1

25m2-------------

Example 15

1

23-----

12 2 2××---------------------

18---



Negative powers of quotients

Proof:

Consider =

= 1 ÷

= 1 ×

=

and =

=

= 1 ÷

= 1 ×

=

=

=

= 1 ÷

= 1 ×

=

and =

= 1 ÷

= 1 ×

=

=

23---

-1 123---

----------

23---

32---

32---

45---

-2 1

45---

2-----------

11625------

----------

1625------

2516------

52

42-----

54---

2

= and = ab---

-1 ba--- a

b---

-n ba---

n

ab---

-1 1 ab

--------------

ab---

ba---

ba---

ab---

-n 1

ab---

n-----------

an

bn-----

bn

an-----

bn

an-----

ba---

n

Example 16

Simplify the following and evaluate if possible.

a b c

Solution

a =

= 1

b =

=

= 2

c =

=

=

45---

-1 35---

-2 2a

b2------

-3

45---

-1 54---

14---

35---

-2 53---

2

259------

79---

2a

b2------

-3 b2

2a------

3

b2( )3

2a( )3-------------

b6

8a3--------


238


1

Express using positive indices:

a

5

−

2

b

3

−

7

c

4

−

1

d

8

−

2

e

10

−

4

f

m

−

1

g

h

−

3

h

w

−

2

i

20

−

4

j

(

−

11)

−

1

k

k

−

8

l

c

−

6

2

Express using positive indices:

a 4d−1 b 3x−5 c 2d−3 d 4m−2 e ab−2

f m2n−4 g wy−2 h 4ac−1 i 3p−2 j 15kw−4

k 12y2m−3 l a−4m2 m d −3y3 n 4xy−3 o v−1m−2

3 Write each of the following using positive indices.a (2m) −1 b (xy) −1 c (4h) −2 d (5k) −3

e (3h) −2 f (4k) −3 g (2c) −4 h (8y) −1

4 Evaluate the following, leaving your answers as fractions.a 3−2 b 4−3 c 6−1 d 7−2

e 11−1 f 2−5 g 4−2 h 10−2

5 Express using negative indices:

a b c d e

f g h i j

k l m n o

p q r s t

6 Evaluate:

a b c d

e f g h

7 Simplify the following and evaluate if possible.

a b c d

e f g h

i j k l

8 Simplify each of the following, using positive indices.a y5 × y−2 b e−3 × e7 c m × m−1 d n6 × n−5 e 4g3 × 3g−1

f 5a−2 × 6a3 g 5x−2 × 2x h 30e−3 × 2e−1 i 8p−1 ÷ 2p2 j 8q ÷ 2q−2

k 2r 2 ÷ 8r −1 l 2t −2 ÷ 8t −1 m (h−1)4 n (b)−3 o (5x−1)2

1m---- 1

w---- 1

8--- 1

9--- 1

22-----

1

n4----- 1

34----- 1

10-3--------- 1

e3----- 1

t2----

2a--- 4

t2---- 2

w5------ 5

d--- 1

2y------

17e------ 1

3a2-------- 5

3m4---------- 1

8p3--------- 2

3k6--------

13---

-1 14---

-2 23---

-2 25---

-3

23---

-1 34---

-1 110------

-5 54---

-1

4w----

-1 mn----

-1 14---

-1 45---

-1

k3---

-1 x3---

-2 a2

4-----

-3

43---

-2

−

2d5

------ -2

h 2

m

3 ------

5 − a2c3

4----------

-3

5d2

p3---------

-3


Example 14

Example 15

Example 16

CAS 7-04

Negative indices

SkillBuilder 11-14

The fifth index law


I ND I C ES

239

CHAPTER 7

9 Simplify the following and express your answers in positive index form.a x3y4 × x−3y−5 b p−4q−1 × 5p2q−3 c (m2n3)−2

d w3p5 ÷ w5p3 e m2n3 ÷ m−5n−1 f 4a3bc2 × −2a−5b−3c−2

g 8xy3 ÷ 4x2y7 h (6m4)−2 × 9m−3 i p2q × p−3q−1 ÷ p4q3

j 8a3h−1 ÷ −4ah ÷ a2h3 k (a2k2)−3 × (a−1k2)−2 l 4x−3y−1 ÷ 8xy3 × 5x−1

m 4r 4t −3 × 5r −5t4 n -15ab−2 ÷ 5a−1b−3 ÷ −6ab7 o (d−3h−1)−1 ÷ −4d3h2

p (2v3w−2)5 ÷ 8v2w−7 q 81a−3e−4 ÷ (3a2e−1)4 r (c−1d −3)3 ÷ (c −1d2)−4

10 Evaluate the following, leaving your answers as fractions.a 23 × 2−4 b (32)−3 ÷ (3−3)3 c 5−1 ÷ 2−1

d 3−2 ÷ 2−1 × 6 e (4−2)2 ÷ (2−2)3 f 32 × (3−2)2

Squaring a number ending in 5, 1 or 9

Squaring a number ending in 5

The square of a number ending in 5 always ends in 25. For example, 352 = 325, and 1052 = 11 025.

A simple calculation trick requires three steps:Step 1: Delete the 5 from the number.Step 2: Multiply the remaining number by the next consecutive number.Step 3: Write ‘25’ at the end of the product.

1 Examine these examples:a 352

Deleting the 5 from 35 leaves just 3.Multiplying 3 by the next consecutive number: 3 × 4 = 12Writing ‘25’ at the end: 1225352 = 1225

b 1052

Deleting the 5 from 105 leaves 10.10 × 11 = 11011 0251052 = 11 025

2 Now calculate these:a 252 b 552 c 452 d 852

e 1152 f 7.52 g 952 h 1952

i 1.52 j 652 k 1552 l 2452


The square of a number ending in 1 always ends in 1. For example, 412 = 1681, and 712 = 5041.

A simple calculation trick requires three steps:Step 1: Subtract 1 to round down to the nearest 10 and make a new number.Step 2: Square the new number.Step 3: Add the new number and the next consecutive number to the square.

Skillbank 7 SkillTest 7-01

Squaring a number ending

in 5, 1 or 9




Round down to 40.Squaring 40: 402 = 1600Adding 40 and 41 to 1600: 1600 + 40 + 41 = 1681412 = 1681

b 712

702 = 49004900 + 70 + 71 = 5041712 = 5041


e 5.12 f 612 g 2012 h 1.12


The square of a number ending in 9 also ends in 1. For example, 292 = 841, and 992 = 9801.

A simple calculation trick requires three steps:Step 1: Add 1 to round up to the nearest 10 and make a new number.Step 2: Square the new number.Step 3: Subtract the new number and the previous consecutive number from the square.


Rounding up gives 30.Squaring 30: 302 = 900Subtracting 30 and 29 from 900: 900 − 30 − 29 = 841292 = 841

b 992

1002 = 10 00010 000 − 100 − 99 = 9801992 = 9801


e 1092 f 4.92 g 792 h 11.92

Note: By combining and adapting the methods for squaring numbers ending in 5, 1 and 9, it is also possible to square a number ending in 4 or 6.

Bonus trick: Squaring a two-digit number beginning with 1

This calculation trick requires three steps:Step 1: Double the units digit and add 10.Step 2: Multiply by 10.Step 3: Add the square of the units digit.


Doubling the units digit and adding 10: 2 × 7 + 10 = 24Multiplying by 10: 24 × 10 = 240Adding the square of the units digit: 240 + 72 = 240 + 49 = 289172 = 289



b 142

2 × 4 + 10 = 1818 × 10 = 180180 + 42 = 180 + 16 = 196142 = 196

8 Now calculate these:a 122 b 132 c 182

d 192 e 112 f 1.62

Working mathematicallyReasoning and reflecting: Fractions as powers(Spreadsheet optional)

1 Copy and complete this table of square numbers and their square roots.

2 Use your calculator to evaluate . (25 1 2 )Now evaluate:

a b c d

3 Look at your calculator answers. Compare them with your answers to Question 1.

Write down what you notice. Predict the values of and .

4 What have you learnt about the fractional power ?

5 Repeat this investigation for fractional cube numbers and their cube roots. Copy and complete this table.

6 Use your calculator to evaluate . (8 1 3 )

Square number Square root

1 1

4 2

9 3

� �

100 10

Cube number Cube root

1 1

8 2

27 3

� �

1000 10

2512---

^ ( abc--- ) =

3612---

6412---

8112---

10012---

1212---

14412---

12---

813---

^ ( abc--- ) =

07_NC_Maths_9_Stages_5.2/5.3 1 Friday, February 6, 2004 2:17 PM


The fractional indexThe square root of a number is the value that, when squared, gives the number. For example,

= 8 because 82 = 8 × 8 = 64.

The cube root of a number is the value that, when cubed, gives the number. For example,

= 5 because 53 = 5 × 5 × 5 = 125.

Now evaluate:

a b c d

7 What do you notice about your answers to Questions 5 and 6?

8 Explain what the fractional power means.

2713---

6413---

51213---

100013---

13---

Calculating square rootsFor computer spreadsheets

Step 1: Set up your spreadsheet as shown.

Step 2: Copy the formulas in cells C7, D6 and E6 to row 34.

Step 3: Print your spreadsheet and paste it in your workbook.

1 Use your spreadsheet to find:a b c d

2 Use your spreadsheet to find:

a b c d

3 a Compare your answers to Questions 1 and 2. Write what you notice.

b Predict the values of and .

4 • In cell F5, insert the heading Number to the power 1/3.• In cell F6, enter the formula =C6^(1/3).• Copy cell F6 down to row 34.

5 Suggest a meaning for:

a b

C D E

5 Number Square root of a number Number to power of 1/2

6 1 =SQRT(C6) =C6^(1/2)

7 =C6+1

8

�

34

9 13 25 29

912---

1312---

2512---

2912---

3612---

6412---

n12---

n13---

Using technology

Spreadsheet 7-02

Calculating square roots

SkillBuilder 11-15

Simplifying fractions

( )

64

3( )

1253



Consider × = = 41

= 4But × = 2 × 2

= 4

∴ =

Proof:

× = = a1

= a

But × = a

∴ =

We also have that × × = = 81

= 8

But × × = 2 × 2 × 2 ( = 2 because 2 × 2 × 2 = 8)

= 8

∴ =

Proof:

× × = = a1

= aBut × × = a

∴ =

412---

412---

412--- 1

2---+

4 4

412---

4

Any number raised to the power of is the square root of that number:

=

12---

a12---

a

a12---

a12---

a12--- 1

2---+

a a

a12---

a

813---

813---

813---

813--- 1

3--- 1

3---+ +

83 83 83 83

813---

83

Any number raised to the power of is the cube root of that number:

=

13---

a13---

a3

a13---

a13---

a13---

a13--- 1

3--- 1

3---+ +

a3 a3 a3

a13---

a3

Example 17

Write each of the following with a fractional index.a b5 113



Solution

a = b =

Write with a fractional index:a b

Solution

a = b =

Evaluate:

a b

Solution

a = = 20 (because 202 = 400)

(Calculator steps: 400 )

b = = 5 (because 53 = 125)

(Calculator steps: 125 )

5 512---

113 1113---

Example 18

g k3

g g12---

k3 k13---

Example 19

40012---

12513---

40012---

400

√– =

12513---

1253

√–3 =

1 Write the following, using fractional indices.

a b c d

e f g h

2 Write the following, using fractional indices.

a b c d

e f g h

3 Evaluate the following.

a b c d

e f g (−64) h

i (−8) j (−729) k l

4 Write the following, using either or .

a b c d

e f g h

5 Evaluate, correct to 2 decimal places:

a b c d

8 35 103 153

20 5123 100 3 72

m w3 8k3 ab

9y3 xy 18 f 10mn3

6412---

34313---

100013---

62512---

0.0412---

0.12513--- 1

3---

102412---

13--- 1

3---

19612---

90012---

3

3712---

813---

d12---

2012---

4p( )13---

100h( )13---

3c7( )12---

w13---

413---

812---

1003 1000


Example 18

Example 19

CAS 7-05

Fractional indices



e f g h

6 Simplify the following.

a × b × c x × d ×

e × f × g h

i j ÷ k 25a ÷ l ÷ 3a

7 Simplify:

a b c d

e f g h

-503 111112---

3.6 -0.008( )13---

n12---

n12---

f14---

f14---

x13---

2m

35---

3m

25---

5g

23---

2g

13---

2h

35---

7g

35---

p2q4( )12---

9a6b6( )12---

8m3n6( )13---

40k13---

8k13---

5a12---

27a23---

m20( )310------

b12( )23---

e8( )34---

p5q15( )45---

25q10( )32---

32h10( )25---

8m3n12( )23---

16x8y4( )54---

Calculating square rootsBefore the advent of calculators and computers, square roots were often calculated manually

in the classroom. Follow the working given below for .

∴ = 239

Use this method to find and . Check your answers using a calculator.

57 121

ROOT (Answer line)

Step 1: Group the digits of 57 121 in pairs from the right.

Step 2: Find the largest square (4) that is less than 5. Subtract it from 5 and write its square root (2) on the answer line above the 5.

Step 3: Bring down the next pair of digits, the 71.

Step 4: Double the number (2) you have in the answer line, to give 4. Write the 4 on the left of the 171. Now, trying 41 × 1, 42 × 2, 43 × 3, 44 × 4, …, find the largest product that is equal to or less than 171. This turns out to be 43 × 3 = 129. Write in 129 and subtract it from 171, leaving 42.

Step 5: Write the 3 in the answer line, above the 71.

Step 6: Bring down the next pair of digits, the 21.

Step 7: Double the number (23) you now have in the answer line, to give 46. Write the 46 on the left of the 4221. Now, by trying 461 × 1, 462 × 2, 463 × 3, …, find the largest product that is equal to or less than 4221. This turns out to be 469 × 9 = 4221. Write in the 4221 and subtract it from 4221, leaving a zero remainder.

Step 8: Write the 9 in the answer line, above the 21.

2 3 95 71 21

− 443 1 71

− 1 29469 42 21

− 42 210

57 121

68 121 173 056

Just for the record



The fractional indices and

Consider =

=

and = = 21

= 2

An iterative process to find the square rootIn the past, before calculators were developed, mathematicians used a process called iteration to give an approximate answer for the square root. One formula for finding an approximation

for is

xn + 1 = , where x0 is a first guess.

Example

Find a good approximation to , correct to two decimal places.

Solution

Guess: x0 = 20

Iteration 1: x1 = = 22.5

Iteration 2: x2 = = 22.361 111 11

Iteration 3: x3 = = 22.360 679 78

This process can continue forever, increasing your accuracy each time.

Hence = 22.36, correct to two decimal places.

The second decimal place does not change for any further iterations.

1 Plan and trial an efficient keystroke sequence to enable you to do this iteration on your calculator. Write down your preferred sequence.

2 a Guess .

b Evaluate to three decimal places, using the iterative formula.

c Evaluate on your calculator to check your answer.

3 a Guess .

b Evaluate to four decimal places, using the iterative formula.

c Evaluate on your calculator to check your answer.

4 Design and trial a spreadsheet to perform this iterative process.

M

12--- xn

Mxn-----+

500

12--- 20 500

20---------+

12--- 22.5 500

22.5----------+

12--- 22.36 500

22.36-------------+

500

300

300

300

71

71

71

Using technology

Spreadsheet 7-03

Square roots (iteration)

1n--- m

n----

25( )15---

2 2 2 2 2××××( )15---

3215---

25( )15---

25 1

5---×



∴ = 2 = (since 2 × 2 × 2 × 2 × 2 = 32)

∴ = ( is read as ‘the fifth root of 32’)

Also =

and =

=

=

= × ×

=

∴ = =

or =

=

=

3215---

325

3215---

325 325

25( )35---

3235---

25( )35---

25( )15--- 3×

25( )

15---

3

3215--- 3

325 325 325

325( )3

3235---

325( )332

35

25( )3 1

5---×

25( )3( )15---

323( )15---

3235

= and

=

=

or =

=

a1n---

an

amn----

a1n--- m

an( )m

amn----

am( )1n---

amn

Example 20

Write each of the following with a fractional index.

a b c

Solution

Evaluate each of the following.

a b c d

Solution

a = b =

=

c =

= d2

a = = 4

b =

= 22

= 4

or =

= = 4

m7 a34 d126

m7 m17---

a34 a3( )14---

a34---

d126 d12( )16---

Example 21

6413---

823---

2753---

8-23---

6413---

643 823---

83( )2

823---

823

643


248


Evaluate the following using a calculator (correct to two decimal places).

a b

Solution

a

=

2.11 (4

20 or 20 1 4 )

b = 3.34 (5 3 4 )

Simplify:

a b

Solution

c

=

=

3

5

=

243

d

=

=

=

=

a

=

=

b =

= 9k4

2753---

273( )5

8-23--- 1

823---

-----

1

83( )2

--------------

1

22

-----

14---

Example 22

2014---

534---

2014---

SHIFT √–x = ^ ( abc--- =

534---

^ ( abc--- ) =

Example 23

16r7( )14---

27k6( )23---

16r7( )14---

1614---r

7 14---×

2r74---

27k6( )23---

27

23---

k6 2

3---×

1 Write each of the following with a fractional index.

a b c d

e f g h

2 Evaluate each of the following.

a b c d e

f g h i j

k5 d54 y

186 x164

n53 e

3 1

n4------- 1

a73

----------

432---

6416---

853---

2723---

1005x---

6443---

3632---

8134---

100023---

8-13---


Example 21

CAS 7-05

Fractional indices



Summary of index laws and properties

k l m n o

3 Evaluate each of the following, correct to two decimal places.

a b c d

e f g h

4 Simplify:

a b c d

e f g h

81-14---

64-43---

400-32---

256-34---

3125-45---

1514---

815---

5054---

6-23---

100-34---

1635---

12-32---

925---

16p4( )14---

8m11( )13---

32d( )25---

27m10( )23---

8d6( )53---

64n12( )43---

25q10( )32---

1000e3d6( )23---

Example 22

Example 23

am × an = am + n

am ÷ an = am − n

(am)n = am × n, = anbn, = a0 = 1

a−n = , = , =

= , = , =

= =

ab( )n ab---

n an

bn-----

1

an----- a

b---

-1 ba--- a

b---

-n ba---

n

a12---

a a13---

a3 a

1n---

an

amn----

an( )m amn

1 Simplify the following, leaving your answers in index form.a 24 × 27 b 159 × 152 c 410 ÷ 47 d 75 × 74 × 73

e 98 ÷ 97 f 38 ÷ 35 ÷ 3 g (43)2 h (82)4

i (73)5 × 74 j 220 ÷ (23)4 k 710 ÷ (73)3 l (145)3 ÷ (142)3

2 Simplify:a x3 × x4 b w8 × w9 c m7 ÷ m2 d k3 ÷ ke (m2)4 f (y4)6 g a3 × a7 ÷ a8 h (p2)3 ÷ p5

i t7 × (t2)3 j (d4)4 ÷ d12 k q6 ÷ q4 × q5 l 2b2 × 3b5

m 4d7 × 5d6 n 30c12 ÷ 5c8 o 24e8 ÷ 6e5 × 2e3 p 15m8 ÷ 5m3 ÷ 3m4


a 30 b c 50 d 7 × 201612---

Exercise 7-09

Worksheet7-03Indices

squaresaw

SkillBuilder 11-20–11-24Review of index

laws



e + 40 f 50 + 100 g (72)0 h +

4 Simplify the following.a (3m5)2 b a2w2 × a3w7 c x3y × x2y4

d p3q5 × pq6 e m5n2 × m2n3 f 2c2d5 × 3c3d3

g 4w5m × 5w3m4 h 2 × 3ab3 × 4a2b3 i (2y3)4 × 3y5

j 4v2 ÷ 2v k m7n3 ÷ m5n2 l a13c6 ÷ a5c2

m 24l 5d3 ÷ 6l 2d n o 36g3h4 ÷ 9g2h2 × 2gh4

5 Evaluate:a 74 ÷ 73 b 85 ÷ 82 ÷ 8 c 43 × 47 ÷ 48 d 243 ÷ 243

e 52 ÷ 50 f 33 + 22 g (33)2 ÷ 35 h 87 × 84 ÷ 811

i (−2)3 j (−2)2 k (−5)0 + (−2)0 l 3 + 30

6 Express with a fractional index:

a b c d

e f g h

7 Express each of these using positive indices:a 7−8 b 2−10 c 15−1 d y−3 e (5x)−1

f 9−2 g 10−3 h (ab)−1 i 4y−8 j (3a2) −1

k 10d−1 l p3q−5 m mw−3 n c2e−3 o 8t3m−4

8 Evaluate the following, leaving your answers as fractions.a 2−2 b 3−1 c 20−1 d 5−3

e 8−3 f 10−4 g 6−4 h 3−5

9 Rewrite these fractions, using negative indices:

a b c d

e f g h

10 Evaluate:a 2−3 b (80)2 c d 4−3 × 43

e 124 ÷ 124 f (43)4 g h 58 ÷ 53

i m0 + k0 j (53)0 k 35 ÷ 37 l 63 ÷ 64


a (4a3)2 b (m2n3)5 c d

e f g h

i j k l

m (4k3n2)4 n o p

2713---

813---

412---

15x5y7

5x2y4-----------------

5 d 3y 103

4p3 xy 10003 3m2n3

1

43----- 1

2--- 1

103-------- 1

74-----

1w---- 1

k4----- 1

d7----- 1

33-----

82( )13---

93( )12---

2k5

m--------

4 4y5

3m2----------

3

45a------

-18c3( )

23--- 10

7m-------

-225w5( )

52---

25

d2------

32---

a3b9

c6-----------

23---

32m105 64

y3------

-23---

1

3g2--------

-216x8( )

54--- 2a3

c2--------

4



Scientific notationWhat is scientific notation?Scientific (or standard) notation is a way of expressing very large or very small numbers using powers of 10. Its use originated in the early twentieth century, when scientists needed to describe very large values, such as astronomical distances, and very small values, such as the masses of atoms. It has the form m × 10n, where m is a number between 1 and 10, and n is an integer.

Working mathematicallyReasoning and reflecting: Repeated roots

This is the square root of the square root of 16.

=

= (the fourth root of 16)

= 2

This is the cube root of the square root of 8.

= (the sixth root of 64)

= (the sixth root of 64)

= 2


a b c d

e f g h

i j

2 Rewrite each of the parts in Question 1 using a single fractional power.

16

16 16( )12--- 1

2---

16( )14---

643

643 64( )12--- 1

3---

64( )16---

81 625 10245 10245

2564 2564 1 000 0003 1 000 0003

256 6561

Worksheet7-04Power

calculations

Worksheet7-05

Binary number system

SkillBuilder 11-16

Scientific notation

SkillBuilder 11-17

Scientific notation for

small numbersNumbers written in scientific (or standard) notation are expressed in the form

m × 10n

where m is a number between 1 and 10 and n is an integer.

Example 24

Express in scientific notation:a 2 700 000 000 b 0.004 67 c 5.78



Solutiona 2 700 000 000= 2 700 000 000

= 2.7 × 109

The power is 9 because the decimal point must move 9 places to the left to put the number in scientific notation.

b 0.004 67= 0.004 67= 4.67 × 10−3

The power is −3 because the decimal point must move 3 places to the right to put the number in scientific notation.

c 5.78 = 5.78 × 100

The power is 0 because the decimal point does not need to move.

Note: Large numbers are written with positive powers of 10, while small numbers are written with negative powers of 10.

Express in decimal form:a 2.7 × 104 b 3.56 × 10−2

Solutiona 2.7 × 104 = 2.7000

= 27 000Since the power is 4, the decimal point moves 4 places to the right to convert to decimal form.

b 3.56 × 10−2 = 0.0356= 0.0356

Since the power is −2, the decimal point moves 2 places to the left to convert to decimal form.

Example 25

1 Express each of the following numbers in scientific notation.a The distance from Earth to the sun is 152 000 000 km.b The world’s largest mammal is the blue whale, which can weigh up to 130 000 kg.c The diameter of an oxygen molecule is 0.000 000 29 cm.d The thickness of a human hair is 0.000 08 m.e Light travels at a speed of 300 000 000 m/s.f The nearest star to Earth, excluding the Sun, is Alpha Centauri, which is

40 000 000 000 000 km away.g The mass of a proton is 0.000 000 000 000 000 000 000 002 g.h The thickness of a typical piece of paper is 0.000 12 m.i The small intestine of an adult is approximately 610 cm long.j The diameter of a hydrogen atom is 0.000 000 0001 m.k The diameter of our galaxy, the Milky Way, is 770 000 000 000 000 000 000 m.l A microsecond means 0.000 001 s.m The Andromeda Galaxy is the most remote body visible to the naked eye, at a distance of

2 200 000 light years.


Worksheet 7-06

Scientific notation



2 Express in scientific notation:a 2400 b 786 000 c 55 000 000 d 95e 7.8 f 348 000 000 g 59 670 h 15i 300 000 000 j 80 k 763 l 24.7m 456.3 n 8.007 o 9057.6 p 130.2

3 Express in scientific notation:a 0.035 b 0.000 076 c 0.8 d 0.0713e 0.000 003 f 0.913 g 0.000 007 146 h 0.009i 0.000 001 j 0.89 k 0.000 000 078 l 0.1

4 Express in scientific notation:a 25 000 b 4 400 000 c 185 000 000d 7 e 0.4 f 0.027g 0.000 875 h 6.7 i 20 345 000 000j 0.000 000 000 9 k 73 l 0.06m 0.552 n 2299 o 3 500 000p 563.7 q 0.0001 r 7.03s 270 000 000 t 400.4 u 50

5 Express each of the following in decimal form.a 5.7 × 103 b 5.7 × 102 c 5.7 × 101 d 5.7 × 100

e 5.7 × 10−1 f 5.7 × 10−2 g 5.7 × 10−3 h 8 × 102

i 8 × 101 j 8 × 100 k 8 × 10−1 l 8 × 10−2

6 Express each of the following in decimal form.a 6 × 105 b 7.1 × 103 c 3.02 × 108

d 3.14 × 100 e 6 × 10−5 f 7.1 × 10−3

g 3.02 × 10−8 h 5.9 × 10−10 i 1.1 × 1012

j 4 × 10−4 k 5 × 103 l 4.76 × 10−4

m 8.03 × 10−1 n 6.32 × 104 o 1.6 × 10−2

p 2.2 × 10−7 q 9.0 × 106 r 1.11 × 10−1

7 Express in scientific notation:a two b ninety c seven hundredd four thousand e five million f three tenthsg seven hundredths h five millionths i fifteen hundredthsj fifteen hundred k three hundred thousand l six thousandths

8 Find the missing power:a 57.3 = 5.73 × 10? b 8 = 8 × 10? c 0.000 004 = 4 × 10?

d 17 000 000 000 = 1.7 × 10? e 4.3 × 10? = 430 f 7.5 × 10? = 0.75g 3.152 × 10? = 3.152 h 1.128 × 10? = 0.000 1128 i 9.05 × 10? = 905 000

Example 25

SkillBuilder 11-18

Scientific notation for

other numbers

Worksheet7-07

Scientific notation puzzle

‘Big’ numbersThe numbers 1000 and 1 000 000 have the names thousand and million , but what about the names of numbers such as 1 000 000 000 and 1 000 000 000 000? The table below lists the names of some big numbers and their meanings.

Just for the record



According to the Guinness Book of Records, the largest number for which there is an accepted name is the centillion, first recorded in 1852. It is equal to 10303.

Find the name of the number that is equal to 10100.

Name Numeral

million 106 = 1 000 000

billion 109 = 1 000 000 000

trillion 1012

quadrillion 1015

quintillion 1018

sextillion 1021

septillion 1024

octillion 1027

nonillion 1030

decillion 1033

Working mathematicallyApply strategies and reasoning: On the blink(Work in pairs.)

1 Copy the following table.

2 Get comfortable and relax. Have a partner watch your eyes and count how often you blink in a minute. Do this twice. Record your results in the table and find the average.

3 Repeat the experiment with you observing your partner. Record your results.

Trial 1 Trial 2 Average blinks per minute

name

name

In common with all crabs, the blue swimmer crab (pictured) cannot blink.



Comparing numbers in scientific notation

4 Calculate how often you each blink in the following period. (Write your answers in scientific notation, correct to two significant figures.)a an hour b a waking day of 16 hoursc a year d an average lifetime of 75 years

5 If a blink takes approximately 0.5 seconds, calculate how long you will each have your eyes closed in the following periods.a a minute b an hour c a waking dayd a waking year e an average waking lifetime

Example 26

Two soft-drink manufacturers each claim that their brand is the most popular in the world, based on last year’s sales. MAXI KOKE sold 5.2 × 1010 cans and KOLA FREE sold 7.9 × 109 cans. Who do you think sold the most?

SolutionWe can check our choice by writing each as an ordinary numeral.

MAXI KOKE: 5.2 × 1010 = 52 000 000 000

KOLA FREE: 7.9 × 109 = 7 900 000 000

Since 52 000 000 000 > 7 900 000 000, we can say that MAXI KOKE had the most sales.

Note that because the powers of ten are different, they are more important than the 7.9 or the 5.2 in making comparisons.

To compare numbers in scientific notation, first compare the powers of ten.

If the powers of ten are the same, then compare the numbers between 1 and 10 that are multiplying the powers of ten.

1 Choose the largest number from each of the following pairs.a 6 × 108 or 8 × 108 b 4.8 × 103 or 2.7 × 105

c 8.4 × 100 or 1.3 × 107 d 3.6 × 10−7 or 6.3 × 10−7

e 9.3 × 109 or 7.6 × 109 f 3.5 × 10−6 or 9.3 × 102

g 3.04 × 100 or 3.04 × 10−4 h 4.5 × 10−5 or 3.7 × 10−7

i 2 × 10−15 or 2 × 10−17 j 6.23 × 10−2 or 9.7 × 10−2

2 Write in order:a 6 × 105, 6 × 102, 6 × 103 (smallest to largest)b 3.8 × 109, 7.3 × 109, 5.5 × 109 (largest to smallest)c 3 × 10−4, 3 × 10−6, 3 × 10−5 (lowest to highest)d 4.1 × 10−3, 9.5 × 10−3, 6.4 × 10−3 (highest to lowest)


Spreadsheet 7-04

Comparing scientific notation

Spreadsheet 7-05

Ordering scientific notation



e 3.5 × 100, 5.3 × 104, 4.9 × 10−4 (ascending order)f 2.1 × 10−8, 6.9 × 10−1, 4.3 × 10−4 (descending order)g 5 × 10−9, 6.3 × 102, 8 × 10−4, 9.76 × 10 (descending order)

3 The following table contains the approximate populations and areas of 10 countries.

a List the countries in descending order of population size.b List the countries in ascending order of area.

Country Population Area (km2)

Australia 2.0 × 107 7.69 × 106

Cambodia 1.3 × 107 1.81 × 105

China 1.3 × 109 9.57 × 106

Indonesia 2.2 × 108 1.90 × 106

Japan 1.3 × 108 3.78 × 105

Lebanon 4.0 × 106 1.02 × 104

New Zealand 3.8 × 106 2.72 × 105

South Africa 4.5 × 107 1.22 × 106

Tonga 9.9 × 104 6.49 × 102

Vietnam 8.0 × 107 3.32 × 105

Hair factsThere are about 1.1 × 105 hairs on your head. Each hair grows at the rate of about 1.3 × 10−3 cm per hour. A single hair lasts about six years. Every day you lose between 30 and 60 hairs. Each hair grows from a small depression in the skin called a follicle (a gland). After the hair falls out, the follicle rests for about three to four months before the next hair starts growing. Hair follicles are either oval, flat or round in shape. How straight, wavy or curly your hair is depends on the shape of your hair follicles.

How many hairs are on all the heads in China?

Curly hair ■ flat follicle Wavy hair ● oval follicle Straight hair ● round follicle

Just for the record



Calculations in scientific notationCalculations using the index lawsScientific notation uses powers of ten, so the index laws can be used to evaluate questions that involve numbers in scientific (or standard) notation.

Working mathematicallyCommunicating and reasoning: Calculator displaysScientific notation can be entered and displayed on the calculator.

A calculator display of 3.507 means 3.5 × 107.

1 Enter the following numbers on your calculator (using the key) and then write down the calculator displays.a 4.7 × 109 b 3.56 × 1015 c 6.7 × 10−6

d 4.2 × 10−10 e 2.047 × 10−4 f 9.8 × 1023

Compare your results with those of other students.

2 Write down these calculator displays in scientific notation:

a b c d

3 Dale and Amy were asked to evaluate 716 to two significant figures.Dale wrote down the answer as 3.313, while Amy wrote the answer as 3.3 × 1013. Which answer is correct? Explain.

4 Explain the difference between the numerical expressions 5 × 107 and 57. Compare your work with that of other students in your class.

5 When entering large or small numbers in a spreadsheet on a computer, E-notation is used. The computer display 6.2E+11 means 6.2 × 1011. Write down these computer displays in scientific notation:a 3.5E+18 b 6.2E–7 c 4.29E–13

EXP

2.7 11 4.02-05 8.7509 1.19-12

Example 27

Use index laws to simplify the following. Give your answers in scientific notation.a (3 × 104) × (8 × 107) b (1.2 × 107) ÷ (3 × 103)

c (5 × 104)2 d

Solutiona (3 × 104) × (8 × 107) = (3 × 8) × (104 × 107)

= 24 × 1011

= 2.4 × 1012

b (1.2 × 107) ÷ (3 × 103) =

= 0.4 × 104

= 4 × 103

4 1010×

1.2 107×

3 103×----------------------

1

0.4



Using the calculator with scientific notationThe calculator may also be used to evaluate expressions involving scientific notation. To enter

scientific notation into the calculator, you need to use the key.

c (5 × 104)2 = 52 × (104)2

= 25 × 108

= 2.5 × 109

d =

= 2 × 105

4 1010× 4

12---

1010( )12---

×

EXP

Example 28

1 Enter each of the following into your calculator.a 6.2 × 1012 b 1.35 × 10−3

Solutiona 6.2 × 1012

Enter: 6.2 12

The calculator display will be , which means 6.2 × 1012 (or ).

b 1.35 × 10−3

Enter: 1.35 3

The calculator display will be (or ).

2 Calculate:a (4.25 × 107) × (8.2 × 106) b (1.08 × 10−15) ÷ (3 × 1011) c (4.9 × 107)2

Solutiona Enter: 4.25 7 8.2 6

Calculator display is (or ) .∴ (4.25 × 107) × (8.2 × 106) = 3.485 × 1014

b Enter: 1.08 3 11

Calculator display is (or ) .∴ (1.08 × 10−15) ÷ (3 × 1011) = 3.6 × 10−27

c Enter: 4.9 7

Calculator display is (or ).∴ (4.9 × 107)2 = 2.401 × 1015

Evaluate each of these, giving your answers in scientific notation, correct to three significant figures.a (5.7 × 105) × (3.42 × 107) b (8.2 × 107)3

Solutiona (5.7 × 105) × (3.42 × 107) = 19.494 × 1012

≈ 1.95 × 1013b (8.2 × 107)3 = 5.513 68 × 1023

≈ 5.51 × 1023

EXP

6.212 6.2 × 1012

EXP (–) =1.35-03 1.35 × 10-03

EXP × EXP =3.48514 3.485 × 1014

EXP (–) 15 ÷ EXP =

3.6-27 3.6 × 10-27

EXP x2 =

2.40115 2.401 × 1015

Example 29



Example 30

Estimate each of the following.a (4.7 × 105) × (3.2 × 108) b (8.4 × 1012) ÷ (1.93 × 107)

Solutiona (4.7 × 105) × (3.2 × 108)

≈ (5 × 105) × (3 × 108)= 15 × 1013

= 1.5 × 1014

b (8.4 × 1012) ÷ (1.93 × 107)

≈

= 4 × 105

8 1012×2 107×--------------------

1 Use the index laws to simplify the following. Give your answers in scientific notation.a (2 × 103) × (3 × 105) b (8 × 107) ÷ (4 × 102)

c (2 × 105)3 de (4 × 107) × (6 × 108) f (1 × 108) ÷ (2 × 103)g (4 × 103)5 h (9 × 105) × (8 × 103) ÷ (4 × 102)i (2 × 10−3)2 j (9 × 10−4) ÷ (3 × 108)k (5 × 108) × (2 × 102)3 l (4.2 × 105) ÷ (6 × 10−5)

2 Find the answers to the following in scientific notation. a (8.4 × 107) × (3.4 × 108) b (9.4 × 1012) + (8.3 × 1015)c (4.9 × 10−9) − (3.7 × 10−10) d (15.75 × 10−3) ÷ (5 × 107)

e 24.08 ÷ (8 × 108) f

g (3.2 × 109)2 h

i j (7.6 × 103) × (4.5 × 105) ÷ (3 × 10−8)

3 Evaluate, giving answers in scientific notation, correct to three significant figures: a (5.12 × 105) × (8.3 × 107) b (2.03 × 1035) + (1.23 × 1034)c (7.4 × 1030) − (3.59 × 1029) d (1.076 × 1017) ÷ (2.3 × 1011)

e f (7.5 × 1023) ÷ (3.3 × 10−13)

g (8.17 × 1016)3 h

i (7.05 × 103) ÷ (3.9 × 107) j

4 Estimate each of the following, leaving your answers in scientific notation.a (5.7 × 103) × (2.3 × 105) b (8.4 × 105) × (3.7 × 107)c (9.1 × 1020) ÷ (3.2 × 105) d (1.6 × 108)2

e (7.13 × 1010) × (9.8 × 108) f (1.99 × 1011) ÷ (2.01 × 107)g (5.85 × 104) ÷ (2.05 × 108) h (6.3 × 1012) ÷ (2.9 × 103)

9 1012×

3.969 1019×

8 10-9×3

7.62 109×2 10-4×

-------------------------

6.6 1027×

2.69 1026×3

5.6 104×( ) 3.9 105×( )×2.3 107×( )

------------------------------------------------------------


Example 28

Example 29

Worksheet7-08

Scientific notation problems



5 Manal’s answer to (8.3 × 1015) × (5.125 × 1017) was 4.25 × 1034, correct to three significant figures.a Estimate an answer to the calculation.b Is Manal’s answer correct? Give reasons.

6 Simplify, giving your answers in scientific notation correct to two significant figures:a 595 × 959 b 1000 ÷ 3 c 220 d 6 ÷ 11e 81−1 f 3−10 g 99 h (0.75)−5

7 a The human body consists of approximately 6 × 109 cells, and each cell consists of 6.3 × 109 atoms. Roughly how many atoms are there in a human body? (Express your answer in scientific notation.)

b The Earth is 1.52 × 108 km from the Sun and the speed of light is 3 × 105 km/s. How long does it take for light to travel from the Sun to Earth? Express your answer in:

i seconds ii minutes.

8 Answer the following in scientific notation, correct to two significant figures where necessary.a A telephone directory is 4.5 cm thick. There are 2000 pages in it. Find the thickness, in

millimetres, of one page.b The Sun burns 6 million tonnes of hydrogen a second. Calculate how many tonnes of

hydrogen it burns in a year (that is, 365 days).c Sound travels at approximately 330 metres per second. If Mach 1 is the speed of sound,

how fast is Mach 5? Convert your answer to kilometres per second.d The distance light travels in one year is called a light year. If the speed of light is

approximately 3 × 105 km per second, how far does light travel in one year?e The nearest star (Alpha Centauri) is 4.3 light years away from Earth. How long would it

take a spaceship travelling from Earth at the speed of light to reach the star?f In a science fiction space movie, Warp 1 is the speed of light. If a starship travels at

Warp 9, which is 9 times the speed of light, how fast is this in metres per second?g A thunderstorm is occurring 33 km from where you are standing. Use the speed of light

(3 × 105 km per second) and the speed of sound (330 metres per second) to calculate:i how long the light from the lightning takes to reach youii how long the sound from the thunder takes to reach you.

9 a What is the largest number that can be displayed on your calculator?b What is the smallest?

14---

Working mathematicallyApplying strategies and reasoning: The reward for inventing chess(Or ‘How many grains of wheat on the chessboard?’)

1 On the first square of a chessboard there is 1 grain of wheat. On the second square there are 2 grains of wheat. On the third square there are 4 grains, on the fourth there are 8 grains, and so on.

2 Copy and complete this table.

Number of square on the board 1st 2nd 3rd 4th … 10th

Number of grains of wheat 1 2 4 8 …



3 Look at the number of each square and the number of grains of wheat on it. What is the connection? Explain it in your own words.

4 How many grains of wheat would be on the:a 16th square? b 32nd square? c 64th square?

5 If a single grain of wheat weighs 100 mg, how many tonnes of wheat would there be on the chessboard?

6 If the inventor had asked for 10c coins instead of grains of wheat, how much money would the king have had to pay?

1 If p = 8, q = 4 and r = 25, evaluate each of the following.a p3 b q4 c r2 d p−1

e q−2 f g h rq3

i (rq)3 j k l pq2r

2 Simplify:a 4y × 4y b 3e ÷ 3e c 10x − 1 × 10x + 1 d 6x + 2 ÷ 6x

e (3a)2 f (32)a g 5n × (5n)2 h (8x)2 ÷ 8x

i m × m + m × m j p × p + p × p + p × p k

3 Write the meaning of each of these:

a b c d e

f g h i j

4 Simplify:

a b c d

e f g h

5 Write each of the following in scientific notation.a 94.2 × 109 b 0.52 × 10−3 c 0.004 × 107 d 105 × 10−4

6 Write each of the following in scientific notation.a 6.7 million b 15.7 million c 57.8 thousandd 4 billion e 3.2 billion f 127 million

7 Find one set of values for each of a, m and n that would make each of the following equations true.

a = 2 b = 3 c = 64 d = 125

Present the results of the following activities in a written report of 1 to 2 pages.

r12---

p13---

qp---

-1 qp---

-2

n12---

n12---

n12---

n12---

×+×

1614---

8114---

3215---

12817---

x34---

m25---

k23---

d710------

3215---

y1n---

a8( )12---

16p8 8y93 27y15( )13---

100g20( )12---

64h24( )13---

n123 64h153

amn----

amn----

amn----

amn----

Power plus



8 For how many values of a and b does ab = ba?

9 Investigate the solutions of the equation xn + yn = an for various values of n.

10 The numbers 3, 5, 17, 257 and 65 537 can all be generated by a simple method, using the numbers 1 and 2 only.a What is this method? b What is the next number in the sequence?

11 On average:• the heart beats 70 times a minute• the skin sweats 0.3 L of liquid a day• the mouth eats 400 grams of food per meal• the lungs breathe 0.6 m3 of air an hour

Calculate the total of each of these activities over a lifetime of 70 years (assuming three meals

a day and 365 days in a year). Express your answwers in scientific notation.

12 Investigate the sonic boom that occurs when an aircraft breaks the sound barrier. (Look up the key terms mach 1 and speed of sound.)

14---

Language of mathsbase cube root EXP expanded form

exponent fractional index index indices

integer negative index power radical sign

reciprocal root scientific notation standard notation

square root zero index

1 Choose five words from the list. Use each word in a sentence to show that you understand its meaning.

2 Match each term in Column A with the correct term in Column B.

3 The following mathematical words have different meanings when used in subjects other than mathematics. For each word, write two sentences, one using the word in its mathematical and one in its its non-mathematical sense.a base b index c power

4 Explain the difference between a base and a power.

5 What is scientific notation used for?

6 Fill in the missing letters:a _ o _ e _ b _ x _ _ n _ n _ c _ c _ e _ t _ f _ _d _ n _ e _ e _ e c _ p _ _ _ a _ f b _ s _

A B

negative power EXP key

standard notation reciprocal

exponent square root, cube root

fractional index index

Worksheet 7-09

Indices crossword



Topic overview• Write 10 questions (with solutions) that could be used in a test for this chapter. Include some

questions that you have found difficult to answer.• Swap your questions with another student and check their solutions against yours.• List the sections of work in this chapter that you did not understand. Follow up this work with

a friend or your teacher.• Copy and complete the summary of this topic shown below. Have your overview checked by

your teacher to make sure nothing is missing or incorrect.

7 How many times can you find the word exponent hidden in the puzzle below?E

X XP P P

O O O ON N N N N

E E E E E EN N N N N N N

T T T T T T T T

INDICES

base

indexor

exponentor

power

Scientific notation• 4 × 102 = 400• 3 × 10−4 = 0.0003• 9372 = 9.372 × 103

Zero index• 80 = 1• 2340 = 1• (1.1)0 = 1

Negative powers

• 7−1 =

•−2

= 2 =

17--

34--

43--

169-----

= 1

10---------- 10

-12---

Index laws1 am × an = am + n

2 am ÷ an = am − n

3 (am)n = amn

4 (ab)n = anbn

5 =

6 a0 = 1

ab--

n an

bn----

= , = ab--

-1 ba-- a

b--

-n ba--

n

Fractional powers

• = • = square root = power of cube root = power of 8

12---

2 513---

53

12--- 1

3---

=

= =

a

1n---

an

a

mn----

an( )m

amn



1 Write each of the following in index notation.a 4 × 4 × 4 b 6 × 6 c 10d x × x × x × x e y × y f m × n × n × ng 7 × 4 × 7 × 7 × 4 h a × b × b × a i 3 × a × 3 × a × a

2 Simplify:a y3 × y10 b a × a4 c h8 × h2

d 3p2 × 2p5 e 3q3 × 3q8 f 5m7 × 2m

g 5x5 × 3x3y h 10x2y × 3xy2 i 4ab4 × 5a3b

3 Simplify:a 420 ÷ 44 b x8 ÷ x2 c b12 ÷ bd 10e15 ÷ 5e3 e 20n9 ÷ 4n3 f 24g8 ÷ 3g2

g h i

4 Simplify:a (a2)4 b (y5)5 c (b)3 d (2x3)5

e (5r 2)3 f (4w4)4 g (a2b)2 h (5a2b)2

i (10g)3 j k l (3m3np2)5

5 Simplify:a 990 b (−99)0 c d0

d (−d)0 e (pq)0 f p0q

g 50 + (5m)0 h 15x0 + (15x)0 i 4 × (9p)0

6 Write each of the following with a positive index.a 8−1 b 8−2 c (9m)−1 d m−5 e y−1

f y−2 g y−3 h (3x)−1 i 3x −1

7 Write each of the following using either or .

a b c d e

f g h i


a b c d

9 Evaluate , correct to two decimal places.

Chapter 7 Review

Ex 7-01

Ex 7-02

Ex 7-03

p6q3

p2q2----------- 36a8b3

4a4b----------------- 100x2y4

5xy2--------------------

Ex 7-04

m4

w5------

5

2a7

e--------

4

Ex 7-05

Ex 7-06

Ex 7-07 3

6412---

6413---

-18---

13---

q15( )13---

p10( )12---

2q( )13---

4a4( )12---

xy( )13---

8k( )12---

Ex 7-08

125d15( )43---

16y20( )14---

32x8( )25--- p3

v6-----

23---

Ex 7-08 4035---

Topic testChapter 7



10 The value of is:A 243 B 81 C 9 D 3

11 The value of is:

A −19.2 B 8 C D

12 Simplify:

a b c

d e f

g h i

j k l

13 Evaluate each of the following.

a + b − c ÷

14 Write each of the following in scientific notation.a 55 000 b 0.55 c 250 000d 0.000 25 e 8 f 0.000 000 000 08

15 Write each of the following in decimal form.a 8.1 × 103 b 6 × 107 c 3.075 × 100

d 8.1 × 10−3 e 6 × 10−7 f 3.075 × 10−2

16 Arrange the numbers in each of these sets in ascending order:a 6.8 × 107, 3.5 × 107, 7.5 × 107

b 3 × 103, 9 × 10−8, 4 × 100

c 4.4 × 10−3, 5.7 × 10−7, 3.1 × 10−1

17 Evaluate each of the following, giving your answers in scientific notation correct to two significant figures.a (3.65 × 10−22) × (7.4 × 108) b (1.44 × 1010) ÷ (3.6 × 104)

c (5 × 105)3 d (6.25 × 10−8)

18 Evaluate, giving your answers in scientific notation correct to two significant figures.

a (3 × 10−8)3 ÷ (2.8 × 10−5) b

c (8.4 × 103)−2 ÷ (4.8 × 107) d (5.64 × 1020)

e 9068 ÷ (0.000 35)2 f × (8.1 × 103)2

g h

Ex 7-082743---

Ex 7-0932-35---

−18--- 1

8---

Ex 7-09

16p4( )34---

-64n123 625w16( )34---

32h10( )-25---

-8x9( )53---

-27m9n3( )-23---

1000c6d9( )43---

32a4b6( )25---

256k8m44

625a8( )-34 -27d12h3( )

53---

32b155

Ex 7-09

64-23---

4-12---

-32( )35---

-8( )43---

932---

36-32---

Ex 7-10

Ex 7-10

Ex 7-11

Ex 7-12

12---

Ex 7-12

7 108×( ) 3.4 105×( )×53---

3.6 10-9×3

9.7 105×( ) 1.3 108×( )×

5.75 10-3×( )2------------------------------------------------------------ 1

1.57 108×-----------------------------


chapter 07 new century maths

Documents