Number and algebraIndicesweb2.hunterspt-h.schools.nsw.edu.au/studentshared/MATHEMATICS… · 5...

5Number and algebra

IndicesThe speed of light is about 300 000 000 metres per second.In one year, light travels approximately 9 460 000 000 000 km.Light from the stars travels for many years before it is seen onEarth. Even light from the Sun takes eight minutes to reachthe Earth. Powers or indices provide a way to work easily withvery large and very small numbers.

n Chapter outlineProficiency strands

5-01 Multiplying anddividing terms withthe same base U F R C

5-02 Power of a power U F R C5-03 Powers of products

and quotients U F R C5-04 The zero index U F R C5-05 Negative indices U F R C5-06 Summary of the

index laws U F R C5-07 Significant figures U F R C5-08 Scientific notation U F R C5-09 Scientific notation

on a calculator U F PS R C

nWordbankbase A number that is raised to a power, meaningit is multiplied by itself repeatedly, for example,in 2 5, the base is 2.

index laws Rules for simplifying algebraic expressionsinvolving powers of the same base, for example,a m 4 a n ¼ a m�n.

index notation A way of writing repeated multiplicationusing indices (powers), for example 2 5.

negative power A power that is a negative number, as inthe term 3�2.

scientific notation A shorter way of writing very large orvery small numbers using powers of 10. For example,9 460 000 000 000 in scientific notation is 9.46 3 1012.

significant figures Meaningful digits in a numeral that tell‘how many’. For example, 28 000 000 has two significantfigures: 2 and 8.

Shut

ters

tock

.com

/Dr_

Flas

h

NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um9

9780170193047

n In this chapter you will:• apply index laws to numerical expressions with integer indices• simplify algebraic products and quotients using index laws• express numbers in scientific notation• interpret and use zero and negative indices• (STAGE 5.2) apply index laws to algebraic expressions with integer indices• round numbers to significant figures• interpret, write and order numbers in scientific notation• interpret and use scientific notation on a calculator• solve problems involving scientific notation

SkillCheck

1 For each term:i state the baseii state the indexiii write the expression in words.

a 84 b 48 c h5 d 5h

2 Evaluate each product.a 5 3 5 b 4 3 4 3 4 c 10 3 10 3 10 3 10 d 2 3 2 3 2 3 2 3 2 3 2e 32 f 25 g 63 h 44

3 Express each repeated multiplication in index notation.

a 2 3 2 3 2 3 2 3 2 b 3 3 3 3 3 3 3 3 7 3 7 3 7c 5 3 5 3 5 3 5 3 5 3 5 3 8 3 8 d 10 3 x 3 x 3 x 3 x 3 x

e 6 3 6 3 6 3 k 3 k f x 3 y 3 x 3 y 3 x

g a 3 b 3 b 3 b 3 a h 5 3 n 3 5 3 n 3 n

i q 3 p 3 q 3 p 3 q 3 q

4 Write each term in expanded form.a 93 b 72 c d5 d k2

5 Evaluate each expression.a 42 3 43 b 106 4 102 c ð33Þ2 d 60

e 91 f 55 3 5 g 24 4 2 h ð�8Þ2

6 For each equation, find the missing power.

a 8 ¼ 2h b 81 ¼ 3h c 216 ¼ 6h d 144 ¼ 12h

e 4096 ¼ 4h f 2401 ¼ 7h g 64 ¼ 2h h 625 ¼ 5h

Worksheet

StartUp assignment 5

MAT09NAWK10052

Worksheet

Powers review

MAT09NAWK10053

Skillsheet

Indices

MAT09NASS10016

174 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

5-01Multiplying and dividing terms withthe same base

Consider 54 3 53 ¼ 5 3 5 3 5 3 5ð Þ3 5 3 5 3 5ð Þ¼ 5 3 5 3 5 3 5 3 5 3 5 3 5

¼ 57

) 54 3 53 ¼ 54þ3

¼ 57

Investigation: Multiplying and dividing terms with powers

1 Write each expression in expanded form, then evaluate it.

a i 22 3 23 ii 25 b i 34 3 33 ii 37

c i 43 3 43 ii 46 d i 55 3 53 ii 58

2 What do you notice about each pair of answers in question 1?3 Is it true that 24 3 26 ¼ 210? Give a reason for your answer.4 Determine whether each equation is true (T) or false (F). Justify your answer.

a 25 3 25 ¼ 210 b 63 3 67 ¼ 621

c 43 3 49 ¼ 427 d 35 3 310 ¼ 315

5 Write in words and as a formula the rule for multiplying am and an, two terms with thesame base.

6 Use the rule to copy and complete each equation.

a 54 3 52 ¼ 5… b 45 3 43 ¼ 4… c 105 3 107 ¼…d 93 3 92 ¼… e n3 3 n8 ¼ … f p3 3 p7 ¼…

7 Evaluate each expression.

a i 36 4 33 ii 33 b i 28 4 26 ii 22

c i 58 4 53 ii 55 d i 108 4 104 ii 104

8 What do you notice about each pair of answers in question 7?9 Is it true that 48 4 46 ¼ 42? Give a reason for your answer.

10 Determine whether each equation is true (T) or false (F). Justify your answer.

a 310 4 36 ¼ 34 b 48 4 42 ¼ 44

c 212 4 23 ¼ 24 d 610 4 65 ¼ 65

11 Write in words and as a formula the rule for dividing am and an, two terms with thesame base.

12 Use the rule to copy and complete each equation.

a 26 4 23 ¼ 2… b 108 4 106 ¼ 10… c 37 4 32 ¼d 411 4 46 ¼… e x8 4 x5 ¼… f g12 4 g10 ¼…

Video tutorial

Simplifying with theindex laws

MAT09NAVT00002

1759780170193047


Summary

When multiplying terms with the same base, add the powers

am 3 an ¼ amþn

The rule above is called an index law. Index is another name for power. The plural of index isindices (pronounced ‘in-de-sees’).Proof:

am 3 an ¼ a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}m factors

3 a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}n factors

¼ a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}ðmþ nÞ factors

¼ amþn

Example 1

Simplify each expression, writing the answer in index notation.

a 84 3 85 b 10 3 103 c d3 3 d5

d 4m2 3 3m6 e 3r2t 3 6r4t3

Solutiona 84 3 85 ¼ 84þ5

¼ 89

b 10 3 103 ¼ 101 3 103

¼ 101þ3

¼ 104

c d 3 3 d 5 ¼ d 3þ5

¼ d 8

d 4m 2 3 3m 6 ¼ 4 3 3ð Þ3 m 2 3 m 6� �¼ 12m 2þ6

¼ 12m 8

e 3r 2t 3 6r4t 3 ¼ 3 3 6ð Þ3 r 2 3 r 4� �3 t 1 3 t 3� �

¼ 18r 2þ4t 1þ3

¼ 18r 6t 4

Consider 56 4 54 ¼ 56

54

¼ 5 3 5 3 5 3 5 3 5 3 55 3 5 3 5 3 5

¼ 5 3 5

¼ 52

) 56 4 54 ¼ 56�4

¼ 52

176 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Summary

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

am 4 an ¼ am

an¼ am�n

This is another index law.Proof:

am 4 an ¼ am

an

¼ a 3 a 3 a 3 a 3 a 3 � � � 3 a

a 3 a 3 a 3 � � � 3 a

ðm factors)ðn factors)

¼ a 3 a 3 . . . 3 a ½ðm� nÞ factors]

¼ am�n

Example 2


a 85 4 83 b 108

10c d20 4 d4

d 20w 10 4 5w 2 e 8x 3y 7

24x 2y

Solutiona 85 4 83 ¼ 85�3

¼ 82

b 108

10¼ 108�1

¼ 107

c d 20 4 d 4 ¼ d 20�4

¼ d 16

d 20w 10 4 5w 2 ¼4 20w 10

1 5w 2

¼ 4w 10�2

¼ 4w 8

e 8x 3y 7

24x 2y 1 ¼1 8x 3�2y 7�1

3 24

¼ xy 6

3

Exercise 5-01 Multiplying and dividing terms withthe same base

1 Which expression is equal to 512 3 53? Select the correct answer A, B, C or D.

A 59 B 515 C 2515 D 2536

2 Simplify each expression, writing the answer in index notation.

a 103 3 102 b 2 3 24 c 32 3 35

d 74 3 7 e 8 3 83 3 84 f 54 3 5 3 54

g 6 3 62 3 63 3 64 h 44 3 44 3 44 i 34 3 30 3 37

j x 3 x4 k g4 3 g4 l w7 3 w

m b3 3 b10 n p10 3 p10 o r 3 r

p y 3 y3 3 y2 q m3 3 m 3 m4 r n8 3 n2

See Example 1

1779780170193047



A 1005 B 1004 C 104 D 105

4 Simplify each expression.

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

d h3 3 5h8 e 3q 3 8q8 f 2a2 3 5a5

g 5n8t 3 6n8t4 h 2ab3 3 15ab i 3e4g3 3 e6g2

j 8p4m5 3 4p3m5 k 16qr8 3 3q7 l 9u3v 3 6uv2w8


A 54 B 59 C 14 D 19


a 107 4 105 b 85 4 8 c 2015 4 205

d 58

52 e 912

93 f 227

23

g 74 4 73 h 220

2i 114 4 114

j p15 4 p10 k n7 4 n l w24 4 w6

m h 20

h 4 n y 8

y 2 o a 12

a 4

p b16 4 b15 q w 25

wr m16 4 m16

7 Which expression is equal to104 4 10? Select the correct answer A, B, C or D.

A 104 B 14 C 13 D 103


a 10y15 4 5y3 b 20w9 4 4w3 c 24r8 4 3r

d 30x 4

x 3 e 10m 10

2mf 4g 12

8g 6

g 14d4h10 4 7hd2 h 15x6y8 4 15xy4 i 6e25d40 4 18e5d4

j 12q 5t 4

16q 4t 3 k 45a 10b 8

5a 5 l 36pq 3r 5

24qr

Investigation: Powers of powers

1 Write each expression in expanded form, then evaluate it.

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?3 Is it true that: (27)3 ¼ 221? Give a reason for your answer.4 Determine whether each equation is true (T) or false (F). Justify your answer.

a (35)3 ¼ 315 b (23)2 ¼ 25 c (210)4 ¼ 214

d (42)5 ¼ 410 e (33)6 ¼ 318 f (52)4 ¼ 56

See Example 2

178 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

5-02 Power of a powerConsider 53� �4 ¼ 53 3 53 3 53 3 53

¼ 5 3 5 3 5ð Þ3 5 3 5 3 5ð Þ3 5 3 5 3 5ð Þ3 5 3 5 3 5ð Þ¼ 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5

¼ 512

) ð53Þ4 ¼ 53 3 4

¼ 512

Summary

When raising a term with a power to another power, multiply the powers:

ðamÞn ¼ am 3 n

Proof:

ðamÞn ¼ am 3 am 3 � � � 3 am|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n factors

¼ a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}m factors

3 a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}m factors

3 ::: 3 a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}m factors|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

n lots of m factors¼ am 3 n

Example 3


a (85)2 b (d3)5 c (2g)4 d (5v4)3 e (�n)6 f (�3t4)3

Solutiona 85� �2 ¼ 85 3 2

¼ 810

b d 3� �5 ¼ d 3 3 5

¼ d 15

c 2gð Þ4 ¼ 24 3 g 4

¼ 16g 4

d 5v 4� �3 ¼ 53 3 v 4� �3

¼ 125 3 v 4 3 3

¼ 125v 12

e �nð Þ6 ¼ �1ð Þ6 3 n 6

¼ 1 3 n 6

¼ n 6

f �3t 4� �3 ¼ �3ð Þ3 3 t 4� �3

¼ �27 3 t 4 3 3

¼ �27t 12

5 Write in words and as a formula the rule for raising am to a power of n, that is, (am)n.6 Use the rule to copy and complete each equation.

a (37)2 ¼ 3… b (52)6 ¼ 5… c (45)2 ¼ 4…

d (a3)4 ¼ a… e (83)7 ¼… f (k4)6 ¼…

Video tutorial


MAT09NAVT00002

Puzzle sheet

Indices puzzle

MAT09NAPS10054

1799780170193047


Exercise 5-02 Power of a power1 Which expression is equal to (103)3 ? Select the correct answer A, B, C or D.

A 303 B 100 C 109 D 106


a (43)2 b (52)8 c (33)4

d (27)4 e (21)2 f (9)3

g (100)2 h (64)5 i (53)5

j (e2)4 k (t5)5 l (y3)7

m (c1)5 n (m7)5 o (y4)4

p (h0)6 q (q6)3 r (w4)1

s (2x)10 t (5n3)8 u (4d3)3

v (�k5)9 w (�d3)4 x (2a8)8

3 Which expression is equal to (�3)5? Select the correct answer A, B, C or D.

A �36 B �35 C 35 D �15


a (2d3)4 b (5m3)2 c (4y5)2

d (3x2)4 e (5u6)5 f (2w5)3

g (10d5)4 h (3e)3 i (2b4)1

j (6d6)2 k (3f 4)5 l (2c3)10

m (�2r)4 n (�5t)3 o (�3m3)2

p (�y3)12 q (�x)3 r (�m3)10

s (�4w5)4 t (�3f )5 u (�3p2)3

v (�3h5)4 w (�10k)2 x (�8y3)1

5-03 Powers of products and quotientsConsider 2 3 5ð Þ3 ¼ 2 3 5ð Þ3 2 3 5ð Þ3 2 3 5ð Þ

¼ 2 3 2 3 2 3 5 3 5 3 5

¼ 23 3 53

) ð2 3 5Þ3 ¼ 23 3 53

Summary

When raising a product of terms to a power, raise each term to that power:

ðabÞn ¼ anbn

See Example 3

Video tutorial


MAT09NAVT00002

Homework sheet

Indices 1

MAT09NAHS10005

180 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Proof:

ðabÞn ¼ ab 3 ab 3 � � � 3 ab|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n factors

¼ a 3 a 3 � � � 3 a|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}n factors

3 b 3 b 3 � � � 3 b|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}n factors

¼ anbn

Example 4

Simplify each expression.

a (�2gh2)5 b (p3q4)2

Solutiona �2gh2� �5 ¼ �2ð Þ5 3 g 5 3 h 2� �5

¼ �32 3 g 5 3 h 2 3 5

¼ �32g 5h 10

b p 3q 4� �2 ¼ p 3� �23 q 4� �2

¼ p 3 3 2 3 q 4 3 2

¼ p 6q 8

Consider 58

� �6

¼ 58

358

358

358

358

358

¼ 5 3 5 3 5 3 5 3 5 3 58 3 8 3 8 3 8 3 8 3 8

¼ 56

86

)58

� �6

¼ 56

86

Summary

When raising a quotient of terms to a power, raise each term to that power:

a

b

� �n¼ an

bn

Proof:a

b

� �n¼ a

b3

a

b3 � � � 3

a

b|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}n factors

¼ a 3 a 3 � � � 3 a ðn factorsÞb 3 b 3 � � � 3 b ðn factorsÞ

¼ an

bn

1819780170193047


Example 5


a 7cd

� �2b 4k 2

5

� �3

Solution

a 7c

d

� �2

¼ 7cð Þ2

d 2

¼ 72c 2

d 2

¼ 49c 2

d 2

b 4k 2

5

� �3

¼ ð4k 2Þ3

53

¼ 43ðk 2Þ3

125

¼ 64k 6

125

Exercise 5-03 Powers of products and quotients1 Which expression is equal to ð�4 3 5Þ2? Select the correct answer A, B, C or D.

A 16 3 25 B �16 3 25 C �8 3 10 D 8 3 10


a (ab)3 b (x2y)5 c (l3m5)6

d (6dp2)4 e (�8k4y5)2 f (3m2n)5

g (ek3)3 h (�w3x4)7 i (�8d3y5)2

j (4b2c3)4 k (�3a3d)3 l (2p2q3)4

3 Which expression is equal to � 34

� �3? Select the correct answer A, B, C or D.

A � 912

B 912

C 2764

D � 2764


a 67

� �2b m

2

� �3c 5

x

� �4

d 2n 5

p

� �8

e w 2

t 3

� �5

f m 2

4n

� �4

g � 23

� �4h � 5h

6

� �3i 7k 4

10

� �2

j 3rt 2

� �2k a 2b

d 5

� �4

l � 23c 2

� �5


a (2x10y15)3 3 5x2y3 b (2x10y15 3 5x2y3)3 c 18q5r8 4 (3qr2)2

d (18q5r8 4 3qr2)2 e 3a 5x 6

ax

� �3

f 3a 5x 6

ðaxÞ4

g (4p3h10)2 3 2p2h9 h (4p3h10)2 4 2p2h9 i (4p3h10 3 2p2h9)2

See Example 4

See Example 5

Worked solutions

Powers of productsand quotients

MAT09NAWS10023

182 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

5-04 The zero index

Consider 53 4 53 ¼ 53

53

¼ 1 Any number divided by itself equals 1.

But also 53 4 53 ¼ 53�3

¼ 50

) 50 ¼ 1:

Summary

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

a0 ¼ 1

Investigation: The power of zero

What is the value of a number raised to a power of 0, for example, 20?1 Copy and complete each table of decreasing powers. Notice the pattern in your answers.

a Power of 2 Number25 3224 1623

22

21

20

b Power of 3 Number35 24334

33

32

31

30

2 Simplify each expression in index notation.

a 34 3 30 b 52 3 50 c 20 3 27

d 70 3 73 e 45 3 40 f 50 3 57

g 25 4 20 h 35 4 30 i 42 4 40

j 93 4 90 k 56 4 50 l 84 4 80

3 Any number will remain unchanged when multiplied by what?4 Any number will remain unchanged when divided by what?5 What is the answer when any number is raised to the power of 0, that is, a0? Justify your

answer.

1839780170193047


Proof:am 4 am ¼ 1 Any number divided by itself equals 1.But also am 4 am ¼ am�m

¼ a 0

) a 0 ¼ 1:

Example 6


a 110 b (�8)0 c g0

d (3r)0 e 3r0 f �80

Solutiona 110 ¼ 1 b (�8)0 ¼ 1 c g0 ¼ 1

d (3r)0 ¼ 1 e 3r 0 ¼ 3 3 r 0

¼ 3 3 1

¼ 3

f �80 ¼ �1 3 80

¼ �1 3 1

¼ �1

Exercise 5-04 The zero index1 Simplify each expression.

a 20 b (�2)0 c �20 d (�m)0

e �m0 f 4að Þ0 g 23

� �0h 7x0

i �10000 j pþ 3ð Þ0 k p3

� �0l �2b 0

m (9k)0 n (x2y)0 o (xyw)0 p (�ab)0

q (6r)0 r �(6r)0 s �6r0 t 6(�r)0

u (cd)0 v �(7x2)0 w �3(a2b3)0 x (�5v5w4)0


a 70 þ 20 b 70 � 20 c 2m0 þ (2m)0 d 2m0 � (2m)0

e (6a)0 þ 6a0 f (6a)0 � 6x0 g (5y)0 � 4 h (5y)0 � 40

i 30 3 50 j 32 3 50 k 12

� �0þ 1

2y 0 l 1

2

� �0þ 1

2y

� �0

m 2w0 3 3p0 n 12u0 4 3 o (5d0)3 p 8b0 � (3b0)2

q 12p 0

ð2pÞ0r 6n 3 4 2n 3 s 12q 5

36q 5 t ð3x 3Þ3 4 x 9

See Example 6

Worked solutions

The zero index

MAT09NAWS10024

184 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Mental skills 5 Maths without calculators

Adding or multiplying in any order

Numbers can be added or multiplied in any order. We can use this property to make ourcalculations simpler.

1 Study each example.

a 19þ 5þ 5þ 1 ¼ 19þ 1ð Þ þ 5þ 5ð Þ¼ 20þ 10

¼ 30

b 13þ 8þ 20þ 27þ 80 ¼ 13þ 27ð Þ þ 20þ 80ð Þ þ 8

¼ 40þ 100þ 8

¼ 148

c 2 3 36 3 5 ¼ 2 3 5ð Þ3 36

¼ 10 3 36

¼ 360

d 25 3 11 3 4 3 7 ¼ 25 3 4ð Þ3 11 3 7ð Þ¼ 100 3 77

¼ 7700

2 Now evaluate each sum.

a 45 þ 16 þ 45 þ 4 þ 7 b 38 þ 600 þ 50 þ 12 þ 40c 18 þ 91 þ 9 þ 20 d 75 þ 33 þ 7 þ 25e 24 þ 16 þ 80 þ 44 þ 10 f 56 þ 5 þ 20 þ 15 þ 4g 100 þ 36 þ 200 þ 10 þ 90 h 54 þ 27 þ 9 þ 16 þ 3i 70 þ 50 þ 30 þ 25 þ 25 j 32 þ 120 þ 40 þ 80 þ 40

3 Now evaluate each product.

a 8 3 4 3 5 b 50 3 7 3 2 c 3 3 5 3 6d 5 3 11 3 40 e 12 3 2 3 3 f 2 3 4 3 25 3 8g 3 3 20 3 7 3 5 h 6 3 8 3 5 3 2 i 2 3 3 3 2 3 11

Investigation: Negative powers

What is the value of a number raised to a negative power, for example, 2�1 or 2�2?1 Copy and complete each table showing decreasing powers. Notice the pattern in your

answers.

a Power of 2 Number23 822 421

20

2�1

2�2

2�3

b Power of 10 Number103 1000102

101

100

10�1

10�2

10�3

1859780170193047


2 Copy and complete this table showing decreasing powers in expanded form. Notice thepattern in your answers.

a Power of 5 Expanded form33 3 3 3 3 332 3 3 331 330 1

3�113

3�21

3 3 3¼ 1

32

3�31

3 3 3 3 3¼ 1

33

3�4

3�5

b Power of 5 Expanded form53 5 3 5 3 552

51

50

5�1

5�2

5�3

5�4

5�5

3 If 3�2 ¼ 132 and 5�3 ¼ 1

53, then write each negative power in a similar way.

a 4�1 b 7�4 c 2�6

4 Simplify each expression in index notation.

a 104 4 107 b 23 4 28 c 34 4 35

d 52 4 58 e a4 4 a6 f a 4 a4

5 Consider104

107 ¼10 3 10 3 10 3 10

10 3 10 3 10 3 10 3 10 3 10 3 10

¼ 110 3 10 3 10

¼ 1103

But also104

107 ¼ 104�7

¼ 10�3

) 10�3 ¼ 1103

Use the method above to show that:

a 23

28 ¼ 2�5 ¼ 125 b 34

35 ¼ 3�1 ¼ 13

c 52

58 ¼ 5�2 ¼ 156 d a4

a6 ¼ a�2 ¼ 1a2

6 Write in words and as a formula the rule for raising a to a negative power �n, that is, a�n.

186 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Technology Negative powersIn this activity we will discover the pattern for negative powers. We will consider base values from2 to 10 shown in column A and indices (powers) �1, �2 and �3 shown in row 1. On aspreadsheet, the symbol for power is ^ (called a carat, press SHIFT 6). For example, 3�1 isentered as 3^�1.

1 Create a spreadsheet as shown below.

2 We will first examine the power of �1. In cell B4, enter ¼A4^$B$1 to calculate 2�1. $B$1is an absolute cell reference, which ensures that the cell does not change when a formula iscopied. This means that in column B, the power will always refer to cell B1 (�1) only. FillDown from cell B4 to B12.

3 Use Format cells to set column B decimals to Fraction and Up to three digits.

4 Compare your answers in column B with the original values in column A. Can you describethe pattern when a base is raised to a power of �1?

5 Now consider powers of �2. Adapt steps from 1 to 3 for column C. Use Fill Down fromcell C4 to C12.

6 Compare your answers in column C with the original values in column A. Can you describethe pattern when a base is raised to a power of �2?

1879780170193047


7 Now consider powers of �3. Adapt steps for column D. In cell D4, enter the formula¼A4^$D$1.

Note: D12’s fraction is missing as it has 4 digits in the denominator, which the spreadsheetdoesn’t allow for. Can you figure out what the fraction should be?

8 Compare your answers in column D with the original values in column A. Can you describethe pattern when a base is raised to a power of �3?

9 Write a rule for negative powers, given the answers you have found in this activity. Discusswith other students in your class.

5-05 Negative indices

Consider 20 4 23 ¼ 20

23

¼ 123

But also 20 4 23 ¼ 20�3

¼ 2�3

) 2�3 ¼ 123

Summary

A number raised to a negative power gives a fraction (with a numerator of 1):

a�n ¼ 1an

Proof:

a 0 4 an ¼ a 0

an

¼ 1an

But also a 0 4 an ¼ a 0�n

¼ a�n

) a�n ¼ 1an

Example 7

Simplify each expression using a positive index (power).

a 2�1 b 5�3 c g�5

Solutiona 2�1 ¼ 1

21 ¼12

b 5�3 ¼ 153 c g�5 ¼ 1

g 5

Worksheet

Power calculations

MAT09NAWK10057

Video tutorial

Negative indices

MAT09NAVT10010

188 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Example 8

Simplify each expression using a positive index.

a 3n�2 b 3nð Þ�2 c p�2q3 d ab�6

Solutiona 3n�2 ¼ 3 3 n�2

¼ 31

31

n 2

¼ 3n 2

b 3nð Þ�2 ¼ 1

3nð Þ2

¼ 19n 2

c p�2q 3 ¼ 1p 2 3 q 3

¼ q 3

p 2

d ab�6 ¼ a 31

b 6

¼ a

b 6

The reciprocal as a powerConsider 9�1 ¼ 1

9

19

is the reciprocal of 9.

Consider23

� ��1

¼ 123

� �

¼ 1 423

¼ 1 332

¼ 32

¼ 112

Summary

A number raised to a power of �1 gives its reciprocal.

a�1 ¼ 1a

a

b

� ��1¼ b

a

Stage 5.2

Video tutorial

Negative indices

MAT09NAVT10010

1899780170193047


Example 9


a 43

� ��1b y

5

� ��1

Solution

a 43

� ��1¼ 3

4b y

5

� ��1¼ 5

y

Exercise 5-05 Negative indices1 Simplify each expression using a positive index.

a 6�2 b 5�7 c 3�1 d 10�2

e g�5 f z�1 g n�3 h t�2

i a�4 j 5�3 k y�d l r�m

2 Evaluate each expression, giving your answers in fraction form.

a 3�2 b 5�4 c 6�1 d 7�2

e 25�1 f 2�7 g 4�3 h 10�6

i 2�10 j 3�3 k 6�2 l 9�4

3 Write each expression using a negative index.

a 1n 2 b 1

nc 1

83 d 18

e 1105 f 2

a 4 g 13

h � 1b

i 6a

j 4t 2 k � 2

w 5 l 5d 3

4 Simplify each expression using positive indices.

a 5h�1 b 2b�5 c 3e�3 d 4n�2

e pb�2 f r2s�4 g w�2y h d�3y3

i (2m)�1 j (xy)�1 k (4h)�2 l (5k)�3

m 3m3p�2 n 15k�1w�4 o 12x�2y�3 p 12x�2y3

q (3h)�2 r (4k)�3 s (2c)�4 t (8y)�1

u 4pq�3 v 4p�1q�3 w vm�2 x v�1m�2


a 27

� ��1b 8

5

� ��1c 9

10

� ��1d 3

2

� ��1

e � 34

� ��1f 5

2

� ��1g x

3

� ��1h 5

a

� ��1

i �m2

� ��1j 5r

4

� ��1k 2

3z

� ��1l 1

v

� ��1

Stage 5.2

See Example 7

Stage 5.2

See Example 8

Worked solutions

Negative indices

MAT09NAWS10025

See Example 9

190 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

5-06 Summary of the index laws

Summary

am 3 an ¼ amþn a0 ¼ 1

am 4 an ¼ am

an¼ am�n a�1 ¼ 1

a

ðamÞn ¼ am 3 n a�n ¼ 1an

ðabÞn ¼ anbn ab

� ��1¼ b

a

ab

� �n¼ an

bn

Exercise 5-06 Summary of the index laws1 Simplify each expression, writing the answer

in index notation.

a 34 3 37 b 210 4 27 c (44)2

d 98 4 9 e (83)4 f 58 3 5

g 65 3 64 3 63 h 38 4 35 3 3 i (23)5 3 24

j 420

45ð Þ2k (82)3 4 8 l 103ð Þ5

102ð Þ7


a a4 3 a3 b t8 3 t c n8 4 n2

d p3 4 p e (w2)4 f (g3)6

g 2b2 3 3b5 h 4d7 3 5d6 i 30c 12

5c 8

j (5b4)4 k 24m6 4 8m4 l (3a)2

3 Evaluate each expression.

a 40 b (�4)0 c 7 3 20 d (7 3 2)0

e (�2)3 f (�3)2 g (52)2 h 24 3 23

i (72)0 j 45 4 42 k 42 4 45 l 103 4 103

m 52 4 50 n 10�2 4 102 o 12

� �0p 10�2 3 102

4 Evaluate each expression, giving your answers in fraction form.

a 5�2 b 2�5 c 20�1 d 10�3

Worksheet

Index laws review

MAT09NAWK10055

Puzzle sheet

Indices squaresaw

MAT09NAPS10056

Homework sheet

Indices 2

MAT09NAHS10006

1919780170193047



a (3mn3)2 b 8a2w2 3 5a3w7 c (4a2b5)4 d 20p 3q 8

5p 2q 6

e 45

� �3f 6c2d0 g 48u 5v 4

16uvh 3x

10

� �2

i (�4n2t)3 j � 23

� �4k 7x 2y 6

35x 5y 3 l p 5

9y

� �2

m (2p3q2)5 n 17n

� �3o �2n0 p ða

2bÞ4 3 a 3

b 5

6 Simplify each expression using a positive index.

a 8�7 b 3�5 c y�1 d x�3

e (5b)�2 f 5b�2 g (ab)�1 h ab�1

i 4t�8 j (11t)�3 k p3q�5 i mw�3

m 8u�3v�4 n �2r6y�5 o 10e�1f 3 p 12

k�4n 7


a 74

� ��1b 5

2

� ��1c 2

3

� ��1d 1

7

� ��1

e r8

� ��1f 1

10p

� ��1

g 6yz

� ��1

h 25a

� ��1


a 143 b 1

2c 1

104 d 192

e 1k

f 9k 4 g �1

x 7 h 5p 3


a q5 3 q�2 b d�3 3 d7 c m�6 4 m5 d t 4 t�1

e 5g3 3 6g�1 f 8a�2 3 3a3 g 7x�2 3 4x h 64p�1

16p 2

i 48q 4 3q�2 j 5t 3

10t�1 k 2(b�1)4 l (3h)�2

5-07 Significant figuresA way of rounding a number is to give the most relevant or important digits of the number. Forexample, a crowd of 47 321 people can be written as 47 000, which is rounded to the nearestthousand, or to two significant figures.

Stage 5.2

NSW

Worksheet

Significant figures

MAT09NAWK10059

192 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

The first significant figure in a number is the first non-zero digit. For example, the significantfigures are shown in bold in this table:

Number First significant digit Number of significant digits47 321 4 547 000 4 20.000 159 2 1 40.000 2 2 1

• When rounding to significant figures, start counting from the first digit that is not 0.• If it is a large number, you may need to insert 0s at the end as placeholders.• Zeros at the end of a whole number or at the beginning of a decimal are not significant: they

are necessary placeholders.• Zeros between significant figures or at the end of a decimal are significant. For example, the

significant figures are shown in bold in this table.

Number First significant digit Number of significant digits809 000 8 30.020 70 2 4

Example 10

State the number of significant figures in each number.

a 63.70 b 0.003 05 c 7600

Solutiona The zero after 7 is significant.

[ 63.70 has four significant figures.

b The first significant figure is 3, and the zero between 3 and 5 is significant.

[ 0.003 05 has three significant figures.

c The zeros after 6 are not significant.

[ 7600 has two significant figures.

Example 11

Round each number to three significant figures.

a 56.357 b 9.249 c 548 307

Solutiona 56.357 � 56.4 b 9.249 � 9.25 c 548 307 � 548 000

The zeros here are notsignificant, but they areplaceholders that arenecessary for showing theplace values of the 5, 4 and 8.

1939780170193047


Example 12

Write each number correct to one significant figure.

a 0.007 39 b 0.025 c 0.963

Solutiona 0.007 39 � 0.007 b 0.025 � 0.03 c 0.963 � 1

Exercise 5-07 Significant figures

1 State the number of significant figures in each number.

a 457 b 0.23 c 15 000d 4.0004 e 0.0005 f 5000g 0.002 07 h 89 072 i 0.040j 76 000 000 k 0.000 328 l 169.320

2 Round each number to three significant figures.

a 37.609 b 9435 c 168.39d 2.813 e 15.99 f 60 522g 1 769 000 h 385 764 i 10.2717

3 Write each number correct to two significant figures.

a 0.0637 b 0.903 c 0.084 55d 0.000 158 e 0.007 625 f 0.038 71g 0.2795 h 0.018 944 i 0.3145

4 What is 45 067 853 rounded to 3 significant figures? Select the correct answer A, B, C or D.

A 45 167 853 B 45 100 000 C 45 067 900 D 45 070 000

5 What is 0.005 605 0 rounded to 2 significant figures? Select the correct answer A, B, C or D.

A 0.01 B 0.010 000 0 C 0.0056 D 0.005 600 0

6 Round each number to one significant figure.

a 9.478 b 57.12 c 0.0367d 0.007 66 e 0.5067 f 10 675g 1856.78 h 0.000 28 i 56 239 400

7 A company makes a profit of $35 754 125.a Round the profit to the nearest million and state the number of significant figures in the

answer.

b Round the profit to the nearest ten million and state the number of significant figures inthe answer.

8 Australia’s population in 2010 was 21 387 000. To how many significant figures has thisnumber been written?

9 A total of 21 558 people attended a local football match. Express this number to threesignificant figures.

The zeros at the beginning ofa decimal are not significant:they are placeholders.

See Example 10

See Example 11

See Example 12

194 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

10 Evaluate each expression, correct to the number of significant figures shown in the brackets.

a 45.6 3 8.7 � 2.75 3 78.32 (2) b 15.5 � 9.87 4 0.24 þ 8.43 3 2.4 (1)

c (63.73 � 27.89) 4 5.82 (3) d 63:25þ 76:0355:89� 89:24

(4)

e 9:732þ 2:76512:27 3 15:8

(1) f 78.91 4 (23.6 þ 94.7) (2)

g 10:941

þ 2530:0076

(3) hffiffiffiffiffiffiffiffiffi84:3p

3 0:0715 (4)

Just for the record Big numbersThe table below lists the names of some big numbers and their meanings.

Name Numeralmillion 106 ¼ 1 000 000billion 109 ¼ 1 000 000 000trillion 1012

quadrillion 1015

quintillion 1018

sextillion 1021

septillion 1024

octillion 1027

nonillion 1030

decillion 1033

According to the Guinness Book of Records, the largest number for which there is anaccepted name is the centillion, first recorded in 1852. It is equal to 10303.What special name for the number 10100?

5-08 Scientific notationScientific notation is a short way of writing very large or very small numbers using powers of 10. Itwas invented 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.

Worksheet

Scientific notation 1

MAT09NAWK00012

Worksheet

Scientific notation 2

MAT09NAWK00019

Worksheet

Scientific notationpuzzle

MAT09NAWK10060

Technology worksheet

Excel worksheet:Scientific notation

MAT09NACT00019

Technology worksheet

Excel spreadsheet:Scientific notation

MAT09NACT00004

Shut

ters

tock

.com

/val

dis

torm

s

1959780170193047


Summary

Numbers written in scientific notation are expressed in the form

m 3 10n

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

Example 13

Express each number in scientific notation.

a 764 000 000 000 b 6000 c 0.0008 d 0.000 000 472

Solutiona Use the significant figures in the number to write a value between 1 and 10: 7.64

Count how many places the decimal point moves to the right to make 764 000 000 000.11 places

764 000 000 000

[ 764 000 000 000 ¼ 7.64 3 1011

b Use the significant figures in the number to write a value between 1 and 10: 6

Count how many places the decimal point moves to the right to make 6000.3 places

6000

[ 6000 ¼ 6 3 103

c Use the significant figures in the number to write a value between 1 and 10: 8

Count how many places the decimal point moves to the left to make 0.0008.4 places

0.0008

[ 0.0008 ¼ 8 3 10�4

Note that small numbers are written with negative powers of 10.

d Use the significant figures in the number to write a value between 1 and 10: 4.72Count the number of places the decimal point moves to the left to make 0.000 000 472.7 places

0.000 000 472

0.000 000 472 ¼ 4.72 3 10�7

Video tutorial

Scientific notation

MAT09NAVT10011

or count the number ofplaces after the first significantfigure, 7

or count the number ofplace after the first significantfigure, 6

or count the number ofdecimal places to the firstsignificant figure, 8

or count the number ofdecimal places to the firstsignificant figure, 4

196 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

Example 14

Express each number in decimal form.

a 2.7 3 104 b 3.56 3 10�2

Solutiona 2.7 × 104 = 2.7000

= 27000Move the decimal point 4 places to the right.

b 3.56 × 10–2 = 0.0356= 0.0356

Move the decimal point 2 places to the left.

Example 15

a Which number is the larger: 3.65 3 1012 or 8.1 3 1012?b Write these numbers in ascending order: 4.3 3 106, 2.8 3 107, 1.9 3 107

SolutionTo compare numbers in scientific notation, first compare the powers of ten.If the powers of ten are the same, then compare the decimal parts.

a The powers of ten are the same. Compare the decimal parts: 8.1 > 3.65.

[ The larger number is 8.1 3 1012

b Compare the powers of ten: 106 < 107.Then compare the two numbers with 107: 1.9 < 2.8.[ The numbers in ascending order are 4.3 3 106, 1.9 3 107, 2.8 3 107.

Exercise 5-08 Scientific notation1 Express each number 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 3 000 000 000 j 80 k 763 l 10m 0.035 n 0.000 076 o 0.8 p 0.0713q 0.000 003 r 0.913 s 0.000 007 146 t 0.009u 0.000 000 1 v 0.000 89 w 0.000 000 078 x 0.1

2 Express each measurement in scientific notation.a The world’s largest mammal is the blue whale,

which can weigh up to 130 000 kg.

b The diameter of an oxygen molecule is0.000 000 29 cm.

c The thickness of a human hair is 0.000 08 m.

d Light travels at a speed of 300 000 000 m/s.

See Example 13

Cor

bis/

�D

enis

Scot

t

1979780170193047


e The nearest star to Earth, excluding the Sun, is Alpha Centauri, which is40 000 000 000 000 km away.

f The thickness of a typical piece of paper is 0.000 12 m.

g The small intestine of an adult is approximately 610 cm long.

h The diameter of a hydrogen atom is 0.000 000 0001 m.

i The diameter of our galaxy, the Milky Way, is 770 000 000 000 000 000 000 m.

j A microsecond means 0.000 001 s.

k The Andromeda Galaxy is the most remote body visible to the naked eye, at a distanceof 2 200 000 light years away.

3 Express each number in decimal form.

a 6 3 105 b 7.1 3 103 c 3.02 3 108

d 3.14 3 100 e 6 3 10�5 f 7.1 3 10�3

g 3.02 3 10�8 h 5.9 3 10�10 i 1.1 3 1012

j 4 3 10�4 k 5 3 103 l 4.76 3 10�4

m 8.03 3 10�1 n 6.32 3 104 o 1.6 3 10�2

p 2.2 3 10�7 q 9.0 3 106 r 1.11 3 10�1

4 For each pair of numbers, write the larger one.

a 3 3 105 or 4 3 105 b 8.4 3 105 or 2.7 3 106

c 8.4 3 100 or 1.3 3 107 d 3.6 3 10�7 or 6.3 3 10�7

e 9.3 3 109 or 7.6 3 109 f 3.5 3 10�6 or 9.3 3 102

g 3.04 3 100 or 3.04 3 10�4 h 4.5 3 10�5 or 3.7 3 10�7

i 2 3 10�15 or 2 3 10�17 j 6.23 3 10�5 or 9.7 3 10�5

5 Write each set of numbers in ascending order.a 3.8 3 109, 7.3 3 109, 5.5 3 109

b 2.2 3 10�4, 5.8 3 10�6, 7 3 10�4

c 3.5 3 100, 5.3 3 102, 4.9 3 102

6 Write each set of numbers in descending order.a 6 3 105, 2.9 3 102, 1 3 102

b 1.2 3 10�9, 6.3 3 102, 8.1 3 10�4

c 4.1 3 10�1, 9.5 3 10�1, 6.4 3 10�3

Scie

nce

phot

oL

ibra

ry/R

ober

tG

endl

er

See Example 14

See Example 15

198 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

5-09 Scientific notation on a calculatorTo enter a number in scientific notation on a calculator, use the or key.

Example 16

Evaluate each expression using scientific notation.

a (4.25 3 107) 3 (8.2 3 106)b (1.08 3 10�15) 4 (3 3 1011)c (4.9 3 107)2

Solutiona Enter 4.25 7 × 8.2 6 = (4.25 3 107) 3 (8.2 3 106) ¼ 3.485 3 1014

b Enter 1.08 − 15 ÷ 3 11 = (1.08 3 10�15) 4 (3 3 1011)¼ 3.6 3 10�27

c Enter 4.9 7 = (4.9 3 107)2 ¼ 2.401 3 1015

Example 17

Estimate the value of each expression in scientific notation, then evaluate it correct to threesignificant figures.

a 9:2 3 109

2:7 3 105 b ð8:5 3 104Þ3 ð6:3 3 107Þ c ð6:08 3 103Þ2

SolutionEstimate Calculated answer

a 9:2 3 109

2:7 3 105 �9 3 109

3 3 105

¼ 93

3109

105

¼ 3 3 104

9:2 3 109

2:7 3 105 ¼ 34 074:074 07

� 34 000

¼ 3:4 3 104

b ð8:5 3 104Þ3 ð6:3 3 107Þ � ð9 3 104Þ3 ð6 3 107Þ¼ ð9 3 6Þ3 ð104 3 107Þ¼ 54 3 1011

¼ 5:4 3 10 3 1011

¼ 5:4 3 1012

ð8:5 3 104Þ3 ð6:3 3 107Þ¼ 5:355 3 1012

� 5:36 3 1012

c ð6:08 3 105Þ3 � ð6 3 105Þ3

¼ 63 3 ð105Þ3

¼ 216 3 1015

¼ 2:16 3 102 3 1015

¼ 2:16 3 1017

ð6:08 3 105Þ3 ¼ 2:24755 . . . 3 1017

� 2:25 3 1017

Worksheet

Scientific notationproblems

MAT09NAWK10061

Homework sheet

Indices 3

MAT09NAHS10007

Homework sheet

Indices revision

MAT09NAHS10008

Puzzle sheet

Scientific notation:accomplishing great

things

MAT09NAPS00005

Note that with scientificnotation on a calculator, thereis no need to enter brackets

( ) around thenumbers.

1999780170193047


Exercise 5-09 Scientific notation on a calculator1 Evaluate each expression using scientific notation.

a (2 3 103) 3 (3 3 105) b (8 3 107) 4 (4 3 102) c (2 3 105)3

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 3 1012p

e (4 3 107) 3 (6 3 108) f (1 3 108) 4 (2 3 103)

g (4 3 103)5 h 24.08 4 (8 3 106) iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3:969 3 1019p

j (2 3 105)�2 kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 3 10�93p

l 7:62 3 109

2 3 10�4

2 Estimate the value of each expression in scientific notation, then evaluate correct to threesignificant figures.

a (5.7 3 103) 3 (2.3 3 105) b (8 3 105) 3 (3.7 3 107)c (9.1 3 1020) 4 (3.2 3 105) d (1.2 3 108)2

e (7.13 3 1010) 3 (9.8 3 108) f (1.9 3 1011) 4 (2.1 3 107)g (5.85 3 104)3 h (6 3 1012) 4 (2.8 3 103)

3 The human body consists of approximately 6 3 109 cells, and each cell consists of 6.3 3 109

atoms. Roughly how many atoms are there in a human body?

4 A telephone book 4.5 cm thick has 2000 pages. Find the thickness of one page, in millimetres

in scientific notation.

5 Evaluate each expression in scientific notation, correct to two significant figures.

a (7.4 3 1030) � (3.59 3 1029) b (1.076 3 1017) þ (2.3 3 1016) cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6:6 3 1027p

d (7.5 3 1023) 4 (3.3 3 10�13) e (8.17 3 1016)3 fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:69 3 1045p

g (7.05 3 103) 4 (3.9 3 107) h 5:6 3 104ð Þ3 3:9 3 105ð Þ2:3 3 107ð Þ i 1595 3 1959

j 520 k 801�1 l 3�10

m 99 n (0.7)�5

Express the answers for questions 6 to 10 in scientific notation correct to two significant figures ifnecessary.

6 The Earth is 1.50 3 108 km from the Sun and the speed of light is 3 3 105 km/s. How longdoes it take for light to travel from the Sun to Earth? Express your answer in:

a seconds b minutes.

7 The Sun burns 6 million tonnes of hydrogen a second. Calculate how many tonnes ofhydrogen it burns in a year (that is, 365.25 days).

See Example 16

See Example 17

Worked solutions

Scientific notation on acalculator

MAT09NAWS10028

Scie

nce

Phot

oL

ibra

ry/P

R.J

.Ber

nard

/CN

RI

200 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

8 Sound travels at approximately 330 metres per second. If Mach 1 is the speed of sound, howfast is Mach 5? Convert your answer to kilometres per second.

9 The distance light travels in one year is called a light year. If the speed of light is approximately3 3 105 km per second, how far does light travel in a leap year?

10 A thunderstorm is occurring 30 km from where you are standing. Use the speed of light(3 3 105 km per second) and the speed of sound (330 metres per second) to calculate inseconds:

a how long the light from the lightning takes to reach you

b how long the sound from the thunder takes to reach you.

11 a What is the largest number that can be displayed on your calculator?

b What is the smallest number that can be displayed?

Investigation: A lifetime of heartbeats

How many times does your heart beat in an average lifetime of 80 years?1 Work in pairs and copy this table.

Name Trial 1 Trial 2 Average beats per minute

2 Use two fingers to measure your pulse. Have your partner time you for a minute. Do thistwice, record your results in the table and find the average.

3 Repeat Step 2 for your partner.4 Calculate how many times your heart (and your partner’s heart) beats in the following

periods. Write your answers in scientific notation correct to two significant figures.a an hour b a day c a weekd a year (use 365.25 days) e an average lifetime of 80 years

iSto

ckph

oto/

clin

tspe

ncer

2019780170193047


Just for the record Hairy numbers

Straight hair• round follicle

Wavy hair oval follicle

Curly hairflat follicle

There are about 110 000 hairs on your head. Each hair grows at the rate of about 1.3 3 10�3 cmper 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 thehair falls out, the follicle rests for about three to four months before the next hair startsgrowing. Hair follicles are either oval, flat or round in shape. How straight, wavy or curlyyour hair is depends on the shape of your hair follicles.

How many hairs are on all the heads in China if its population is approximately 1.435 3 109?Answer in both scientific notation and decimal notation.

Power plus

1 If u ¼ 2, v ¼ 4 and w ¼ 5, evaluate each expression.

a 5w�2 b �u�3 c vu3 d (wu)3

e 2vw 2

� ��1f w

u

� ��2g u0(vw)�2 h (uvw)�2


a 5a 3 5a b 4x 4 4x c 2 y�1 3 2 y þ 1 d 10bþ3 4 10b

e (3p)3 f (33)p g 8n 3 (8n)2 h ð6tÞ3 4 6t

i p 3 p þ p 3 p j n 3 n þ n 3 n þ n 3 n k q 3 qqþ q

3 Write each number in scientific notation.

a 438.2 3 109 b 0.52 3 10�7 c 0.0004 3 1012

d 2013 3 10�3 e 57.8 thousand f 57.8 thousandthsg 6.7 millionths h 3.2 billion i 3.2 billionths

4 For how many values of a and b does ab ¼ ba?

5 The terms in the pattern 3, 5, 17, 257, 65 537,… can all be generated by a simplemethod, using only the numbers 1 and 2.

a What is this method?b What is the next number in the sequence?

Shut

ters

tock

.com

/with

God

Shut

ters

tock

.com

/Sto

ckL

ite

Shut

ters

tock

.com

/Jas

onSt

itt

Worksheet

Binary number system

MAT09NAWK10058

202 9780170193047

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13

Indices

UmamaheswariG

Text Box

Power plus

Chapter 5 review

n Language of mathsascending base descending estimate

expanded form exponent index index laws

index notation indices negative power placeholder

product power quotient reciprocal

significant figures scientific notation term zero power

1 What does a power of zero mean?

2 Which two words from the list mean ‘power’?

3 What is the or key on a calculator used for?

4 What is the index law for dividing terms with the same base?

5 Which digits in 0.006 701 are significant figures?6 What power is associated with the reciprocal of a term or number?

7 What type of numbers when written in scientific notation have negative powers of 10?

n Topic overview• What was this topic about? What was the main theme?• What content was new and what was revision?• What are the index laws?• 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.• 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 this mind map of the topic, adding detail to its branches and using pictures,symbols and colour where needed. Ask your teacher to check your work.

Zero andnegativeindices

Significantfigures

Power ofa power

Powers ofproducts

and quotients

Multiplying anddividing terms

with the same base

Scientificnotation

Index orpower

Base

INDICES

Puzzle sheet

Indices crossword

MAT09NAPS10062

Quiz

Numbers and indices

MAT09NAQZ00002

Worksheet

Mind map: Indices

MAT09NAWK10063

9780170193047 203


a 103 3 107 b 420 4 44 c a12 4 a2 d h8 3 h2

e 3n3 3 4n f 10d15 4 5d3 g 20m9 4 4m h 3v4w2 3 2v3w5

i 5x5y2 3 3xy j 24t 8h 8

3t 4h 2 k p 6q 10

p 2q 2 l 100a 2b 4

5ab 2


a (22)3 b (k5)5 c (x)4

d (2y3)10 e (5t2)2 f (10g)3

g �2ð Þ5 h �2kð Þ5 i ð�5m 3Þ2


a (ab2)4 b (5x3y2)2 c (4t2)3 d (�4h2g)3

e a7

� �4f (2pqr)5 g 3m

2

� �5h (�3np2)4

i 2a 7

b

� �4

j (4t4u5)3 3 8t2u k b 8y 6

8b 2y

� �3

l 45c6d8 4 (3cd2)2


a 70 b (�7)0 c e0

d (�e)0 e �e0 f g0h

g (gh)0 h 2p3

� �0i 2p

3

0


a 8�3 b 19�2 c x�1 d p�5


a (4m)�1 b (4m)�2 c (5b)�1 d 5b�1

e �2x�4 f 35a

� ��1g c�4d2 h 100

9

� ��1


a 1103 b 1

r 5 c 1r

d 3b


a 96u5 t8 4 24ut4 b (5ab)2 c 4hd3 3 5h3d6

d e5 3 e 3 (e2)3 e (r3)4 4 r2 f (pn2)3 4 (pn3)2

g a 8b 3ca 4b 3c 2 h 60n 4m 3

12n 8mi 144t 4u 3v

16u 9v 4

9 Round each value correct to the number of significant figures shown in the brackets.

a 8.5678 (2) b 15 712 (3) c 476 (1)d 0.007 126 6 (4) e 0.9041 (3) f 301 378 (2)g 4805.28 (3) h 0.000 87 (1) i 67 000 000 (1)

Stage 5.2

See Exercise 5-01

See Exercise 5-02

See Exercise 5-03

See Exercise 5-04

See Exercise 5-05

See Exercise 5-05

See Exercise 5-05

See Exercise 5-06

See Exercise 5-07

9780170193047

Chapter 5 revision

204

10 Express each number in scientific notation.

a 37 000 b 0.61 c 250 000d 0.000 49 e 13 f 0.000 000 000 08

11 Express each number in decimal form.

a 8.1 3 103 b 6 3 107 c 3.075 3 100

d 8.1 3 10�3 e 6 3 10�7 f 3.075 3 10�2

12 Write these numbers in ascending order: 3 3 103, 9.1 3 10�8, 2.4 3 103.

13 Evaluate each expression using scientific notation.

a (3.65 3 1022) 3 (7.4 3 108) b (1.44 3 1010) 4 (3.6 3 104)

c (5 3 105)3 dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6:25 3 10�8p

14 Estimate the value of each expression in scientific notation, then evaluate correct to twosignificant figures.

a (8.9 3 109) 3 (1.1 3 107) b (9.3 3 1015) 3 (4 3 102) c (3.1 3 104)2

See Exercise 5-08

See Exercise 5-08

See Exercise 5-08

See Exercise 5-09

See Exercise 5-09

9780170193047

Chapter 5 revision

205

Number and algebraIndicesweb2.hunterspt-h.schools.nsw.edu.au/studentshared/MATHEMATICS… · 5...

Documents

Transcript of Number and algebraIndicesweb2.hunterspt-h.schools.nsw.edu.au/studentshared/MATHEMATICS… · 5...