number and algebraindicesweb2.hunterspt-h.schools.nsw.edu.au/studentshared/mathematics… · 5...
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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.
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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
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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
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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
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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
Simplify each expression, writing the answer in index notation.
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
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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 53? Select the correct answer A, B, C or D.
A 54 B 59 C 14 D 19
6 Simplify each expression, writing the answer in index notation.
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
8 Simplify each expression.
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
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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
Simplify each expression, writing the answer in index notation.
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
Simplifying with theindex laws
MAT09NAVT00002
Puzzle sheet
Indices puzzle
MAT09NAPS10054
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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
2 Simplify each expression, writing the answer in index notation.
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
4 Simplify each expression.
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
Simplifying with theindex laws
MAT09NAVT00002
Homework sheet
Indices 1
MAT09NAHS10005
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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
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um9
Example 5
Simplify each expression.
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
2 Simplify each expression.
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
4 Simplify each expression.
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
5 Simplify each expression.
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
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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.
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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
Simplify each expression.
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
2 Simplify each expression.
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
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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
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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.
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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$1is an absolute cell reference, which ensures that the cell does not change when a formula iscopied. This means that in column B, the power will always refer to cell B1 (�1) only. FillDown from cell B4 to B12.
3 Use Format cells to set column B decimals to Fraction and Up to three digits.
4 Compare your answers in column B with the original values in column A. Can you describethe pattern when a base is raised to a power of �1?
5 Now consider powers of �2. Adapt steps from 1 to 3 for column C. Use Fill Down fromcell C4 to C12.
6 Compare your answers in column C with the original values in column A. Can you describethe pattern when a base is raised to a power of �2?
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7 Now consider powers of �3. Adapt steps for column D. In cell D4, enter the formula¼A4^$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
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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
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um9
Example 9
Simplify each expression.
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
5 Simplify each expression.
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
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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
2 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
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
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5 Simplify each expression.
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
7 Simplify each expression.
a 74
� ��1b 5
2
� ��1c 2
3
� ��1d 1
7
� ��1
e r8
� ��1f 1
10p
� ��1
g 6yz
� ��1
h 25a
� ��1
8 Write each expression using a negative index.
a 143 b 1
2c 1
104 d 192
e 1k
f 9k 4 g �1
x 7 h 5p 3
9 Simplify each expression.
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
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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NEW CENTURY MATHSfor the A u s t r a l i a n C u r r i c u l um9
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
2 Simplify each expression.
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?
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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 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
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9780170193047 203
1 Simplify each expression, writing the answer in index notation.
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
2 Simplify each expression, writing the answer in index notation.
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
3 Simplify each expression.
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
4 Simplify each expression.
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
6 Simplify each expression using a positive index.
a (4m)�1 b (4m)�2 c (5b)�1 d 5b�1
e �2x�4 f 35a
� ��1g c�4d2 h 100
9
� ��1
7 Write each expression using a negative index.
a 1103 b 1
r 5 c 1r
d 3b
8 Simplify each expression.
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
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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
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