number and algebraindicesweb2.hunterspt-h.schools.nsw.edu.au/studentshared/mathematics… · 5-04...

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 and dividingterms with the 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 C

5-06 Fractional indices1n* U F R C

5-07 Fractional indicesm

n* U F R C

5-08 Summary of the index laws U F R C5-09 Significant figures U F R C5-10 Scientific notation U F R C5-11 Scientific notation on a

calculator U F PS R C

*STAGE 5.3

nWordbankbase A number that is raised to a power, meaning it ismultiplied by itself repeatedly, for example, in 25, the base is 2.

index laws Rules for simplifying algebraic expressions involvingpowers of the same base, for example, am 4 an¼ am�n.

index notation A way of writing repeated multiplicationusing indices (powers), for example 25.

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

power (or index or exponent) The number of times abase appears in a repeated multiplication, for example, in25, the power is 5.

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.

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9780170193085

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

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 5.3) interpret and use fractional 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 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

3 Write each term in expanded form.

a 93 b 72 c d5 d k2

4 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

5 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

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

Indices

5-01Multiplying and dividing terms withthe same base

Consider 5 4 3 5 3 ¼ ð5 3 5 3 5 3 5Þ3 ð5 3 5 3 5Þ¼ 5 3 5 3 5 3 5 3 5 3 5 3 5

¼ 57

) 5 4 3 5 3 ¼ 5 4þ3

¼ 57

Summary

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

am 3 an ¼ amþn

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 the samebase.

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

1719780170193085


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

Proof: a m 3 a n 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 a mþn

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 8 4 3 8 5 ¼ 8 4þ5

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 4m2 3 3m6 ¼ ð4 3 3Þ3 ðm2 3 m6Þ¼ 12m2þ6

¼ 12m8

e 3r2t 3 6r4t 3 ¼ ð3 3 6Þ3 ðr2 3 r4Þ3 ðt 1 3 t3Þ¼ 18r2þ 4t 1þ3

¼ 18r6t 4

Consider 56 4 54 ¼ 56

54

¼ 6 5 3 6 5 3 6 5 3 6 5 3 5 3 56 5 3 6 5 3 6 5 3 6 5

¼ 5 3 5¼ 52

) 56 4 54 ¼ 56�4

¼ 52

Summary

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

a m 4 a n ¼ a m

a n ¼ a m�n

This is another index law.

Proof: a m 4 a n ¼ a m

a n

¼ 6 a 3 6 a 3 6 a 3 a 3 a 3 � � � 3 6 a6 a 3 6 a 3 6 a 3 � � � 3 6 a

ðm factors)ðn factors)

¼ a 3 a 3 � � � 3 a ½ðm� nÞ factors]

¼ a m�n

172 9780170193085

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

Indices

Example 2


a 85 4 83 b 10 8

10 c d20 4 d4

d 20w 10 4 5w 2 e 8x 3y7

24x 2y

Solutiona 8 5 4 83 ¼ 85�3

¼ 82

b 10 8

10 ¼ 108�1

¼ 107

c d 20 4 d 4 ¼ d 20�4

¼ d16

d 20w 10 4 5w 2 ¼4 20w 10

16 5w 2

¼ 4w 10�2

¼ 4w 8

e 8x 3y 7

24x 2y1 ¼16 8x 3�2y 7�1

324¼ xy 6

3

Exercise 5-01 Multiplying and dividing termswith the 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

3 Which expression is equal to 104 3 10? Select the correct answer A, B, C or D.

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

5 Which expression is equal to 512 4 5 3? Select the correct answer A, B, C or D.

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

See Example 1

See Example 2

1739780170193085


g 74 4 73 h 220

2 i 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

w r m16 4 m16

7 Which expression is equal to 10 4 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 10m10

2mf 4g12

8g 6

g 14d 4h10 4 7hd2 h 15x6y8 4 15xy4 i 6e25d40 4 18e5d4

j 12q 5t 4

16q 4t 3 k 45a 10b8

5a 5 l 36pq3r5

24qr

5-02 Power of a powerConsider ð5 3Þ4 ¼ 5 3 3 5 3 3 5 3 3 5 3

¼ ð5 3 5 3 5Þ3 ð5 3 5 3 5Þ3 ð5 3 5 3 5Þ3 ð5 3 5 3 5Þ¼ 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

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

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…

Puzzle sheet

Indices puzzle

MAT09NAPS10054

Video tutorial


MAT09NAVT00002

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

Indices

Summary

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

ða mÞn ¼ a m 3 n

Proof: ða mÞn ¼ a m 3 a m 3 � � � 3 a m|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}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|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}m factors|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

n lots of m factors¼ a m 3 n

Example 3


a (85)2 b (d3)5 c (2g)4

d (5v4)3 e (�n)6 f (�3t4)3

Solutiona ð8 5Þ2 ¼ 85 3 2

¼ 810b ðd 3Þ5 ¼ d 3 3 5

¼ d15c ð2gÞ4 ¼ 24 3 g4

¼ 16g4

d ð5v4Þ3 ¼ 5 3 3 ðv4Þ3

¼ 125 3 v4 3 3

¼ 125v12

e ð�nÞ6 ¼ ð�1Þ6 3 n6

¼ 1 3 n6

¼ n6

f ð�3t 4Þ3 ¼ ð�3Þ3 3 ðt 4Þ3

¼ �27 3 t 4 3 3

¼ �27t 12

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

See Example 3

1759780170193085


5-03 Powers of products and quotientsConsider ð2 3 5Þ3 ¼ ð2 3 5Þ3 ð2 3 5Þ3 ð2 3 5Þ

¼ 2 3 2 3 2 3 5 3 5 3 5

¼ 23 3 5 3

) ð2 3 5Þ3 ¼ 23 3 5 3

Summary

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

ðabÞn ¼ a nbn

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 ð�2gh2Þ5 ¼ ð�2Þ5 3 g 5 3 ðh2Þ5

¼ �32 3 g 5 3 h 2 3 5

¼ �32g 5h10

b ð p 3q 4Þ2 ¼ ð p 3Þ2 3 ðq 4Þ2

¼ p 3 3 2 3 q4 3 2

¼ p6q8

Consider58

� �6¼ 5

8 358 3

58 3

58 3

58 3

58

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

� �6¼ 56

86

Video tutorial


MAT09NAVT0002

Homework sheet

Indices 1

MAT09NAHS10005

176 9780170193085

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

Indices

Summary

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

ab

� �n¼ a n

b n

Proof: ab

� �n¼ a

b3

ab

3 � � � 3ab|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

n factors

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

¼ a n

b n

Example 5


a7cd

� �2

b 4k 2

5

� �3

Solution

a 7cd

� �2¼ 7cð Þ2

d 2

¼ 72c 2

d 2

¼ 49c 2

d 2

b 4k 2

5

� �3

¼ ð4k 2Þ35 3

¼ 43ðk 2Þ3125

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 (ek 3)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 9

12 C 2764

D � 2764


a 67

� �2b m

2

� �3c 5

x

� �4d 2n 5

p

� �8

e w 2

t 3

� �5

f m 2

4n

� �4

g � 23

� �4h � 5h

6

� �3i 7k4

10

� �2

j 3rt 2

� �2k a2b

d 5

� �4

l � 23c 2

� �5

See Example 4

See Example 5

1779780170193085



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

d (18q5r8 4 3qr2)2 e 3a 5x6

ax

� �3

f 3a 5x6

ðaxÞ4

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

5-04 The zero indexConsider 5 3 4 5 3 ¼ 5 3

5 3

¼ 1 Any number divided by itself equals 1.

But also 5 3 4 5 3 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.

Worked solutions

Powers of productsand quotients

MAT09NAWS10023

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

Indices

Proof: am 4 am ¼ 1 Any number divided by itself equals 1.

But also a m 4 a m ¼ a m�m

¼ a0

) a0 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 �2b0

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

2 y0 l 12

� �0þ 1

2 y� �0

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

q 12p0

ð2pÞ0r 6n3 4 2n3 s 12q 5

36q 5 t ð3x 3Þ3 4 x9

See Example 6

Worked solutions

The zero index

MAT09NAWS10024

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Mental skills 5 Maths without calculators

Adding or multiplying in any orderNumbers 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 ð19þ 1Þ þ ð5þ 5Þ¼ 20þ 10¼ 30

b 13þ 8þ 20þ 27þ 80 ¼ ð13þ 27Þ þ ð20þ 80Þ þ 8¼ 40þ 100þ 8¼ 148

c 2 3 36 3 5 ¼ ð2 3 5Þ3 36¼ 10 3 36¼ 360

d 25 3 11 3 4 3 7 ¼ ð25 3 4Þ3 ð11 3 7Þ¼ 100 3 77¼ 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

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Indices

Technology Negative powersIn this activity we will discover the pattern for negative powers. We will consider base valuesfrom 2 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.

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 13�1 1

33�2 1

3 3 3 ¼132

3�3 13 3 3 3 3 ¼

133

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.

1819780170193085


1 Create a spreadsheet as shown below.

2 We will first examine the power of �1. In cell B4, enter 5A4^$B$1 to calculate 2�1.$B$1 is an absolute cell reference, which ensures that the cell does not change whena formula is copied. This means that in column B, the power will always refer to cellB1 (�1) only. Fill Down 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?

7 Now consider powers of �3. Adapt steps for column D. In cell D4, enter the formula5A4^$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?

182 9780170193085

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

Indices

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 1a n

Proof: a0 4 a n ¼ a0

a n

¼ 1a n

But also a0 4 a n ¼ a0�n

¼ a�n

) a�n ¼ 1a n

Example 7

Simplify each expression using a positive index (power).

a 5�3 b 3n�2 c 3nð Þ�2 d p�2q3

Solutiona 5�3 ¼ 1

5 3 b 3n�2 ¼ 3 3 n�2

¼ 31 3

1n 2

¼ 3n 2

c 3nð Þ�2 ¼ 13nð Þ2

¼ 19n 2

d p�2q 3 ¼ 1p 2 3 q 3

¼ q 3

p 2

Worksheet

Power calculations

MAT09NAWK10057

Video tutorial

Negative indices

MAT09NAVT10010

1839780170193085


The reciprocal as a powerConsider 9�1 ¼ 1

919 is the reciprocal of 9.

Consider23

� ��1¼ 1

23

� �¼ 1 4

23

¼ 1 332

¼ 32

¼ 1 12

Summary

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

a�1 ¼ 1a

ab

� ��1¼ b

a

Example 8


a 43

� ��1b y

5

� ��1

Solution

a 43

� ��1¼ 3

4 b y5

� ��1¼ 5

y

Negative powers of quotients

Consider45

� ��2¼ 1

45

� �2

¼ 11625

¼ 1 41625

Stage 5.3

184 9780170193085

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

Indices

¼ 1 32516

¼ 2516

¼ 52

42

¼ 54

� �2

Summary

A number raised to a power of –n gives its reciprocal raised to the power of n.

ab

� ��n¼ b

a

� �n

¼ b n

a n

Proof:ab

� ��n¼ 1

ab

� �n

¼ 1a n

b n

¼ b n

a n

¼ ba

� �n

Example 9


a 43

� ��3b 2 1

2

� ��2c 3a

b4

� ��2

Solutiona 4

3

� ��3¼ 3

4

� �3

¼ 2764

b 2 12

� ��2¼ 5

2

� ��2

¼ 25

� �2

¼ 425

c 3ab4

� ��2¼ b4

3a

� �2

¼ b8

9a2

Stage 5.3

1859780170193085


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

n c 183 d 1

8

e 1105 f 2

a 4 g 13 h � 1

b

i 6a j 4

t 2 k � 2w 5 l 5

d 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


a 14

� ��2b 2

3

� ��2c 1

10

� ��6d 5

2

� ��3

e 43

� ��5f 5

4

� ��4g 2 1

4

� ��2h 1 2

5

� ��3

i k3

� ��2j 3

x

� ��3k a2

4

� ��4

l 43g3

� ��2

m 2d5t

� ��2n h 2

m3

� ��5

o 5d 3

3p4

� ��2

p 3c 3

4a 2

� ��3

See Example 7

Worked solutions

Negative indices

MAT09NAWS10025

See Example 8

Stage 5.3

See Example 9

186 9780170193085

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

Indices

5-06 Fractional indices 1n

We now know the meaning of zero and negative indices, that is, a0, a�1 and a�n.What is the meaning of fractional indices, that is, a

12 and a

1n ?

Consider 2512

� �2¼ 25

12 3 2

¼ 251

¼ 25

Power of a power

butffiffiffiffiffi25p� 2¼ 25

) 2512 ¼

ffiffiffiffiffi25p

¼ 5

Summary

Any number raised to the power of 12 is the square root of that number:

a12 ¼

ffiffiffiap

Proof: a12

� �2¼ a

12 3 2

¼ a1

¼ aBut

ffiffiffiapð Þ2 ¼ a

) a12 ¼

ffiffiffiap

Now consider 2713

� �3¼ 27

13 3 3 Power of a power

¼ 271

¼ 27but

ffiffiffiffiffi273p� 3 ¼ 27

) 2713 ¼

ffiffiffiffiffi273p

¼ 9

Summary

Any number raised to the power of 13 is the cube root of that number:

a13 ¼

ffiffiffia3p

Proof: a13

� �3¼ a

13 3 3

¼ a1

¼ aBut

ffiffiffia3pð Þ3 ¼ a

) a13 ¼ ffiffiffi

a3p

Now consider 3215

� �5¼ 32

15 3 5 Power of a power

¼ 321

¼ 32

Stage 5.3

1879780170193085


If 3215

� �5¼ 32, then 32

15 is called the 5th root of 32, written

ffiffiffiffiffi325p

.

25 ¼ 32

) 3215 ¼

ffiffiffiffiffi325p

¼ 2

Summary

Generally, any number raised to the power of 1n is the nth root of that number:

a1n ¼

ffiffiffianp

Proof: a1n

� �n¼ a

1n 3 n

¼ a1

¼ a

Butffiffiffianpð Þn ¼ a

) a1n ¼

ffiffiffianp

Example 10

Evaluate each expression.

a 90012 b 125

13 c 1024

110

Solutiona 900

12 ¼

ffiffiffiffiffiffiffiffi900p

¼ 30

b 12513 ¼


¼ 5

c 10241

10 ¼ffiffiffiffiffiffiffiffiffiffi102410p

¼ 2

Enter on calculator: 10 3 1024 =

because 210 ¼ 1024

Summary

On a calculator, the nth root key is 3 or , found by pressing the SHIFT or 2ndF keybefore pressing or yx respectively.

Example 11

Write each expression using a fractional index.

affiffiffi8p

bffiffiffiffiffi363p

cffiffiffin4p

dffiffiffiffiffiab7p

Solutiona

ffiffiffi8p¼ 8

12 b

ffiffiffiffiffi363p

¼ 3613 c

ffiffiffin4p¼ n

14 d

ffiffiffiffiffiab7p

¼ ðabÞ17

Stage 5.3

188 9780170193085

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

Indices

Exercise 5-06 Fractional indices1n


a 2512 b 343

13 c 625

12 d 1000

13

e 3215 f ð�0:027Þ

13 g 0:04ð Þ

12 h 64

13

i �8ð Þ13 j �729ð Þ

13 k 256

18 l 3125

15

2 Write each expression using a radical (root) sign.

a 1012 b 12

13 c g

12 d m

14

e 8rð Þ12 f 6hð Þ

16 g 5j8ð Þ

15 h 90ab

19

3 Write each expression using a fractional index.

affiffiffi5p

bffiffiffiffiffi493p

cffiffiffiffiffi20p

dffiffiffiffiffiffiffiffi4005p

effiffiffiffiffi666p

fffiffiffiffiffi644p

gffiffiffiffiffiffiffiffi1448p

hffiffiffiffiffiffiffiffiffiffi100010p

iffiffiffiap

j ffiffiffiq3p k

ffiffiffih7p

lffiffiffiffiw6p

mffiffit5p

n ffiffiffiffiffixyp o

ffiffiffiffiffiffiffiffiffiffi100f4p

pffiffiffiffiffiffiffiffiffi2mn3p

4 Evaluate each expression correct to 2 decimal places.

a 2013 b 215

12 c


dffiffiffiffiffiffiffiffiffiffi2001p

e ð�666Þ13 f

ffiffiffiffiffiffiffiffiffiffi11114p

gffiffiffiffiffiffiffiffiffiffiffi�7545p

hffiffiffiffiffiffiffiffiffiffiffi0:0086p


a b12 3 b

12 b e

13 3 e

13 3 e

13 c y 3 y

15 d m

35 3 m

25

e 2t13 3 5t

23 f �6a

32

� �2g n12m4ð Þ

14 h 16a 2b6

� 12

i 8v 6w 9� 1

3 j 40a1

10 4 8a110 k 35x 4 5x

13 l 36y

34 4 4y

5-07 Fractional indices mn

What is the meaning of fractional indices such as a23 and a

32 ?

Consider 3235 ¼ 32

15

� �3Power of a power

¼ffiffiffiffiffi325p� �3

¼ 23

¼ 8

or consider 3235 ¼ 323� 1

5 Power of a power

¼ffiffiffiffiffiffiffi3235p

¼ffiffiffiffiffiffiffiffiffiffiffiffiffi32 7685p

¼ 8

Summary

amn ¼

ffiffiffianp� m or

ffiffiffiffiffiffia mnp

Stage 5.3

See Example 10

See Example 11

Worked solutions

Fractional indices

MAT09NAWS10026

1899780170193085


Proof: amn ¼ a

1n

� �mor a mð Þ

1n

¼ffiffiffianp� m or


Note: Taking the root first often makes the calculation simpler.

Example 12

Evaluate each expression.

a 823 b 27

43 c 64�

13 d 16�

34

Solutiona 8

23 ¼

ffiffiffi83p� �2

¼ 22

¼ 4

b 2743 ¼

ffiffiffiffiffi273p� �4

¼ 34

¼ 81

c 64�13 ¼ 1

6413

¼ 1ffiffiffiffiffi643p

¼ 14

d 16�34 ¼ 1

1634

¼ 1ffiffiffiffiffi164p� 3

¼ 123

¼ 18

Example 13

Evaluate 30035 correct to two decimal places.

Solution

30035 ¼ 30:63887063 . . .

� 30:64

Enter on calculator: 300 3 5 =

Example 14

Write each expression using a fractional index.

affiffiffiffiffip 34

pb

ffiffiffiffiffib7p

c 1ffiffiffiffiffiq 43

pSolutiona

ffiffiffiffiffip 34

p¼ p 3� 1

4

¼ p34

bffiffiffiffiffib7p

¼ b7� 12

¼ b72

c 1ffiffiffiffiffiq 43

p ¼ 1

q 4ð Þ13

¼ 1

q43

or q�43

Stage 5.3

190 9780170193085

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

Indices

Example 15


affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16r 2ð Þ34

qb

ffiffiffiffiffiffiffiffi27k3p� 2 c 32a 5ð Þ�

35

Solution

affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16r 2ð Þ34

q¼ 16r 2� 3

4

¼ 1634r 2 3 3

4

¼ 8r32

bffiffiffiffiffiffiffiffi27k3p� �2

¼ 27kð Þ23

¼ 2723k

23

¼ 9k23

c 32a 5� �35 ¼ 1

32a 5ð Þ35

¼ 1

3235a 5 3 3

5

¼ 18a3

Exercise 5-07 Fractional indicesm

n


a 432 b 8

53 c 128

57 d 27

23

e 102435 f 64

43 g 32

35 h 81

34

i 100023 j 125

23 k 8�

13 l 81�

14

m 25�32 n 36�

32 o 256�

34 p 3125�

45

q 1024�45 r 400�

32 s 128�

47 t 1024�

710

2 Evaluate each expression correct to two decimal places.

a 1534 b 8�

75 c 50

54 d 6�

23

e 100�34 f 16

35 g 12�

32 h 179

25


affiffiffiffiffig27

pb

ffiffiffiffiffie5p

cffiffiffiffiffiffix186p

d 1ffiffiffiffiffiffiy164

pe

ffiffiffiffiffiffim53p

fffiffiffiffiffiffim35p

g 1ffiffiffiffiffin34p h 1ffiffiffiffiffi

n43p


a 16n4ð Þ34 b 8wð Þ

23 c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32d10ð Þ35

qd

ffiffiffiffiffiffiffiffiffiffiffi64m8p� �3

effiffiffiffiffiffiffiffiffiffi81r 44p� �5

fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi81h12ð Þ34

qg 1ffiffiffiffiffiffiffi

8s63p� �4 h 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1024x155p� �2

i 125b6� �2

3 j 625t 2ð Þ�34 k 49p4q10ð Þ

32 l 1000x 3y 6

� �23

Stage 5.3

See Example 12

See Example 13

See Example 14

See Example 15

1919780170193085


5-08 Summary of the index laws

Summary

am 3 an ¼ amþn a0 ¼ 1

a m 4 a n ¼ a m

a n ¼ a m�n a�1 ¼ 1a

ða mÞn ¼ a m 3 n a�n ¼ 1a n

ðabÞn ¼ a nb n ab

� ��1¼ b

a

ab

� �n¼ a n

b nab

� ��n¼ b

a

� �n

¼ b n

a n

a12 ¼

ffiffiffiap

, a13 ¼

ffiffiffia3p

, a1n ¼

ffiffiffianp

amn ¼

ffiffiffianpð Þm or


Exercise 5-08 Summary of the index laws1 Simplify each expression.

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


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

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

a 5�2 b 2�5 c 20�1 d 10�3


a 1612 b 27

13 þ 40 c 25

52 d 8

13 þ 4

12

e 82ð Þ13 f 93ð Þ

12 g 81

34 h �32ð Þ

35

Worksheet

Index laws review

MAT09NAWK10055

Puzzle sheet

Indices squaresaw

MAT09NAPS10056

Homework sheet

Indices 2

MAT09NAHS10006

Stage 5.3

192 9780170193085

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

Indices


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

5p 2q 6

e 45

� �3f 6c2d0 g 48u5v4

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 a3

b5

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 11t�3 j (11t)�3 k p3q�5 i mw�3

m 8u�3v�4 n �2r6y�5 o 10e�1f 3 p 12 k�4n7


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

2 c 110 4 d 1

92

e 1k

f 9k 4 g �1

x7 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


affiffiffi5p

bffiffiffid3p

cffiffiffiffiffi3yp

dffiffiffiffiffi104p

e ffiffiffip3p� 2 f

ffiffiffiffiffiffiffiffiffiffixyð Þ5

qg

ffiffiffiffiffiffiffiffiffiffiffi5að Þ34

qh

ffiffiffiy6

q� �5


a 45a

� ��3b 8c 3ð Þ

23 c 10

7m

� ��2d 25w5ð Þ

52

e 49d 2

� �32 f a3b9

c6

� �23

gffiffiffiffiffiffiffiffiffiffiffiffiffi625m64p

h 64y 3

� ��23

iffiffiffiffiffiffiffiffiffiffiffiffi32m105p

j 13g 2

� ��2

k 16x8ð Þ54 l 2a3

c 2

� �4

Stage 5.3

Worked solutions

Summary of theindex laws

MAT09NAWS10027

1939780170193085


5-09 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.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 16

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.

NSW

Worksheet

Significant figures

MAT09NAWK10059

194 9780170193085

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

Indices

Example 17

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

Example 18

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 5-09 Significant figures1 State the number of significant figures in each number.

a 457 b 0.23 c 15 000 d 4.0004 e 0.0005 f 5000g 0.002 07 h 89 072 i 0.040 j 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.

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

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

See Example 16

See Example 17

See Example 18

1959780170193085


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.

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 10:941þ

2530:0076

(3) hffiffiffiffiffiffiffiffiffi84:3p

3 0:0715 (4)

Just for the record Big numbers

The 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?

5-10 Scientific notationScientific notation is a short way of writing very large orvery small numbers using powers of 10. It was inventedin the early twentieth century when scientists needed todescribe very large values, such as astronomical distancesand very small values such as the masses of atoms.

Shut

ters

tock

.com

/val

dis

torm

s

Worksheet

Scientific notation 1

MAT09NAWK00012

Worksheet

Scientific notation 2

MAT09NAWK00019

Puzzle sheet

Scientific notationpuzzle

MAT09NAPS10060

196 9780170193085

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

Indices

Summary

Numbers written in scientific notation are expressed in the form

m 3 10 n

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

Example 19

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

0.00080.0008

[ 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

Technology worksheet

Excel worksheet:Scientific notation

MAT09NACT00019

Technology worksheet

Excel spreadsheet:Scientific notation

MAT09NACT00004

Video tutorial

Scientific notation

MAT09NAVT10011

or count the number ofplaces after the first significantfigure, 7

or count the number ofplaces 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

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Example 20

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 21

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 5-10 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 moleculeis 0.000 000 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 19

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e The nearest star to Earth, excluding the Sun, is Alpha Centauri, whichis 40 000 000 000 000 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 distance of2 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

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See Example 20

See Example 21

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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

5-11 Scientific notation on a calculatorTo enter a number in scientific notation on a calculator, use the or key.

Example 22

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 23

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 10 4Þ3 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 3

109

105

¼ 3 3 10 4

9:2 3 109

2:7 3 105 ¼ 34 074:074 07

� 34 000

¼ 3:4 3 10 4

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.

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Estimate Calculated answerb ð8:5 3 10 4Þ3 ð6:3 3 107Þ� ð9 3 10 4Þ3 ð6 3 107Þ¼ ð9 3 6Þ3 ð10 4 3 107Þ¼ 54 3 1011

¼ 5:4 3 10 3 1011

¼ 5:4 3 1012

ð8:5 3 10 4Þ3 ð6:3 3 107Þ ¼ 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

Exercise 5-11 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?

See Example 22

See Example 23

Worked solutions

Scientific notationon a calculator

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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 10 4ð 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).

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 calculatein seconds: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?

Worked solutions

Scientific notationon a calculator

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Indices

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. Eachhair 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 folliclesare either oval, flat or round in shape. How straight, wavy or curly your hair is depends on theshape 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.

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

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Power plus

1 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


affiffiffiffiffiffiffiffiffiffiffiffiffiffi

81pp

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

625pp

cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

256ppq

dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

10245pp

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6561ppq

fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

256p4

pg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 000 0003pp

hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 000 000p3

p3 Find values of a, m and n so that each equation is true.

a amn ¼ 2 b a

mn ¼ 3 c a

mn ¼ 64 d a

mn ¼ 125

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?

Worksheet

Binary number system

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

Indices

Chapter 5 review

n Language of maths

ascending base descending estimate

expanded form exponent fractional power index

index laws index notation indices negative power

power product quotient reciprocal

scientific notation significant figures term zero power

1 What does a power of 12 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 (or print) and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.

Zero andnegativeindices

Index orpower

Significantfigures

Power ofa power

Powers ofproducts

and quotients

Multiplying anddividing terms

with the same base

Scientificnotation

Fractionalindices

Base

INDICES

Puzzle sheet

Indices crossword

MAT09NAPS10062

Quiz

Numbers and indices

MAT09NAQZ00002

Worksheet

Mind map: Indices(Advanced)

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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 5x5y 2 3 3xy

j 24t 8h8

3t 4h 2 k p6q10

p 2q2 l 100a 2b4

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�5m3Þ2


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

d (�4h2g)3 e a7

� �4f (2pqr)5

g 3m2

� �5h (�3np2)4 i 2a7

b

� �4

j (4t4u5)3 3 8t2u k b8y 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

5 Simplify each expression using a positive index.

a 8�3 b 19�2 c x�1 d p�5

e (4m)�1 f (4m)�2 g (5b)�1 h 5b�1

i �2x�4 j 35a

� ��1k c�4d2 l 100

9

� ��1


a 1103 b 1

r 5 c 1r d 3

b

7 Simplify � 83x

� ��2using a positive index.

8 Write each expression using a radical (root) sign.

a q13 b u

12 c 2qð Þ

13 d arð Þ

12


a 6423 b �32ð Þ

35 c 36�

32


a 125d15ð Þ43 b 16y20ð Þ

14 c 32x8ð Þ

25 d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�64n123p

11 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)

See Exercise 5-01

See Exercise 5-02

See Exercise 5-03

See Exercise 5-04

See Exercise 5-05

See Exercise 5-05

Stage 5.3

See Exercise 5-05

See Exercise 5-06

See Exercise 5-07

See Exercise 5-07

See Exercise 5-09

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12 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

13 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

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

15 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 d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6:25 3 10�8p

16 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 5-10

See Exercise 5-10

See Exercise 5-10

See Exercise 5-11

See Exercise 5-09

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