Chapter 07 New Century Maths
Short Description
Descripción: Indices chapter of New Century Math textbook...
Description
07_NC_Maths_9_Stages_5.2/5.3 Page 218 Friday, February 6, 2004 2:17 PM
7
Number/Patterns and algebra
Indices
The speed of light is about 300 000 km/s. In one year, light travels approximately 9 460 000 000 000 km. The light from the stars travels for many years before it is seen on Earth. Powers or indices provide a way to work easily with very large numbers or with very small numbers.
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In this chapter you will: ■ ■ ■ ■ ■ ■ ■ ■
describe and evaluate numbers written in index form, using terms such as ‘base’, ‘power’, ‘index’ and ‘exponent’ develop and use the index laws for multiplying and dividing terms with the same base, and for the power of a base raised to a power develop and use zero and negative indices use fractional indices for square roots and cube roots express and order numbers in scientific notation convert numbers expressed in scientific notation to decimal form enter and read scientific notation on a calculator calculate with numbers expressed in scientific notation.
Wordbank ■ ■
■ ■ ■
base A number that is raised to a power, meaning that it is multiplied by itself repeatedly. For example, in 2 5, the base is 2. power The number of times a base is multiplied by itself. For example, 2 5 means 2 × 2 × 2 × 2 × 2, and is 2 to the power of 5. A power is also called an index or an exponent. index notation or index form Repeated multiplication written in the form a n. For example, 2 × 2 × 2 × 2 × 2 written using index notation is 2 5. negative power A power that is a negative number, as in the expression 3 −2. scientific notation A shorter way of writing very large or very small numbers using powers of 10. For example, 9 460 000 000 000 in scientific notation is 9.46 × 10 12.
Think! The story is that Sissa ben Dahir, who invented chess, was offered any reward he wanted by the Indian King Shirham. Sissa asked for the following: ‘I will have one grain of wheat for the first square of my chessboard, two grains of wheat for the second, four for the third and so on to the sixtyfourth square.’ King Shirham granted his request without thinking! ■ How many grains of wheat would be needed for the 64th square? ■ How many grains of wheat would be needed altogether to meet Sissa’s request? ■ If a grain of wheat weighs 100 mg, how many tonnes of wheat would there be on the chessboard? I NDI CE S
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Start up Worksheet 7-01 Brainstarters 7
1 Evaluate: a 4×4 d 5×5×5 g 4×4×4 2 Evaluate: a 43
b 102
3 Express in index form: a 5×5 d 3×3×3×3×3×3 g a×a×a
b 2×2×2×2×2 e 10 × 10 × 10 × 10 × 10 h 8×8×8×8×8×8
c 3×3 f 7×7 i 12 × 12 × 12 × 12
c 26
e 42
d 34
b 4×4×4×4×4 e y×y h x×x×x×x×x×x
4 Write in expanded form: a 103 b 82 2 f k g w4
c 15 h d5
106
f
c 6×6×6 f m×m×m×m×m i d×d×d×d d 24 i p1
e 31 j c3
5 Evaluate: 400
a
289
b
c
1024
-8
d
d
225
625
e
6 Evaluate: 3
a
8
3
b
27
c
3
3
-216
e
3
1000
f
3
-27
Powers The numbers 2, 4, 8, 16, … are powers of 2. (They can also be written as 21, 22, 23, 24,… .) Similarly, the numbers 3, 9, 27, 81, 243, … are powers of 3.
Working mathematically Reasoning and reflecting: Powers and the power key Numbers expressed as powers of numbers, such as 27, can be easily evaluated using the power key (
^
or
xy
or
y x ) on your calculator.
1 a Evaluate 24 = 2 × 2 × 2 × 2 = ? b Evaluate 24 using the power key on your calculator as follows: 2
^
4
=
(Note: Your answers for parts a and b should be the same.) 2 Use the power key to evaluate each of the following. Compare your answers to those of other students. a 45 b 77 c 34 d 118 3 a Copy the table below into your book and use your calculator to evaluate the first six powers of 4, 5, 6 and 7, and enter them in your table. Compare your results with those of other students in your class.
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Powers of 4 41
Powers of 5
=
51
42 =
=
=
56
Powers of 7
=
71 =
62 =
72 =
61
52 =
46
Powers of 6
=
66
=
76
b Using the results in your table, evaluate: i 81 ii 151 iii 21 c What is the value of a1 (that is, any number to the power of 1)?
=
iv 231
4 a Evaluate the powers of 2 (21, 22, 23, …). What is the largest power of 2 that your calculator can display as a whole number? b Find the largest power of each of the following numbers that your calculator can display as a whole number.
i 3
ii 4
iii 5
iv 6
v 7
Compare your results with those of other students in your class.
Index notation Consider 2 × 2 × 2 × 2 × 2 = 25. 2 × 2 × 2 × 2 × 2 is the expanded or factor form. 25 is the index notation or exponent form. In 25, 5 is called the index or the power or the exponent. The base is 2. 25 is read as ‘2 to the power of 5’ or ‘2 to the 5th’.
Skillsheet 7-01 Indices
25
SkillBuilder 11-01 Introduction to indices
index, power or exponent
base
Example 1
Solution
c a×a×…×a n factors
b m × m × m × m × m = m5 5 factors
c a × a × … × a = an n factors
a 3 × 3 × 3 × 3 = 34 4 factors
b m×m×m×m×m
Express in index form: a 3×3×3×3
Example 2 b p×p×p×p×t×t×t×t×t×t
Express in index form: a 5×5×5×6×6×6×6 c a×a×…×a × b×b×…×b n factors m factors
Solution
a 5 × 5 × 5 × 6 × 6 × 6 × 6 = 53 × 64 3 factors 4 factors b p × p × p × p × t × t × t × t × t × t = p4 × t 6 = p4t 6 4 factors 6 factors
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c a × a × … × a × b × b × … × b = a n × b m = a nb m n factors m factors
Example 3 Express in expanded form: a 35
b 5x 3m 4
Solution
a 35 = 3 × 3 × 3 × 3 × 3
b 5x 3m 4 = 5 × x × x × x × m × m × m × m
Exercise 7-01
Example 1
1 For each of the following: i state the base ii state the index a 37 b 73 c k4
iii write the expression in words. d 4k e an
2 Express in index form: a 5×5×5×5 b 10 × 10 e 9×9×9×9×9 f 12 × 12 × 12
c 8×8×8 g 1×1×1×1
3 Write using index notation: a a×a×a×a d q×q×q×q×q×q Example 2
Example 3
CAS 7-01 Index form
b m×m e p×p×p
d 32 × 32 h 6
c y×y×y×y f w
4 Write these in index notation: a 3×3×2×2×2×2×2 b 3×3×3×3×7×7×7 d 6×6×6×k×k e x×y×x×y×x
c 5×5×5×5×5×5×8×8 f 5×n×5×n×n
5 Write in expanded or factor form: b 103 a 64 4 e 5p f 52p4 2 4 5 i 5pq j ab3 3 4 m mn n 2y3d 2
c g k o
64 × 103 p4q5 ab3c2 42a3m
d h l p
p4 5p4q5 a4bc 2 w 4y 2v 3
6 Evaluate the following. a 24 b e 27 f 4 i 6 j m 52 × 55 n
c g k o
52 132 210 44 × 6 2
d h l p
43 83 35 35 × 53
33 53 73 33 × 10 4
7 Evaluate, correct to three decimal places: a 3.17
b (0.145)2
c
4 − 2--- 5
d (−2.5)7
e (1.1)5
f
4 1 2--- 7
g
5 − 2--- 3
h (0.18)2
8 Find the missing powers in: a 8 = 2? b 81 = 3? ? e 4096 = 2 f 2401 = 7? 9 Evaluate, correct to 2 significant figures: b (−11)5 a 126 3 d (3.1) e (−1.11)2
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c 216 = 6? g 64 = 2?
d 144 = 12? h 625 = 5? c 212 f (7.2)4
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10 If p = 4, q = 3, r = 5, evaluate the following. a p4 b pq3 d pq2r
c (pq)3
e (pqr)2
f
q--- r
3
11 a Evaluate the terms of the pattern (−1)1, (−1)2, (−1)3, (−1)4, (−1)5, (−1)6, … Write down what you observe about the odd and even powers and the sign of the answer. b Without using a calculator, find the value of: i (−1)98 ii (−1)99 c Predict the values of the following. i (−1)n when n is even ii (−1)n when n is odd d Hence evaluate: i (−1)1 + (−1)2 + (−1)3 + (−1)4 + … + (−1)36 ii (−1)1 + (−1)2 + (−1)3 + (−1)4 + … + (−1)37 12 a Evaluate the following. i 62 ii 662 b Predict the values of the following. i (666 666)2
iii 6662
iv 66662
ii (666 666 666)2 Spreadsheet 7-01 Cell growth
Working mathematically Applying strategies and reasoning: Cell growth Use a spreadsheet to help you with this investigation. Over the centuries, millions of people have contracted diseases such as smallpox, typhoid and diphtheria. These diseases start off as a few cells that multiply at an alarming rate until there are too many in the body, causing the person to become ill. In some cases this can be fatal. Suppose one of these diseases grows by the cells splitting into equal parts every 10 seconds; that is, every 10 seconds, the number of cells doubles. Disease A t=0s
1 cell
t = 10 s
2 cells
t = 20 s
4 cells
1 Starting with one cell, calculate the cell population after: a 30 s b 40 s c 1 min e 2 min f 3 min g 4 min
d 1 min 30 s h 4 min 20 s
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2 Starting with one cell, find how long it will take until there are: a 64 cells b 256 cells c over 500 cells d over 1000 cells e over 3000 cells f over 10 000 cells g over 1 000 000 cells h over 40 000 000 cells Further diseases are discovered that multiply at different rates: • Disease B: A single cell divides into three identical cells every 15 seconds. • Disease C: A single cell divides into four identical cells every 20 seconds. • Disease D: A single cell divides into five identical cells every 30 seconds. 3 Expand your spreadsheet to show all four disease strains, and answer Questions 1 and 2 for strains B, C and D. 4 If the diseases all begin from one cell at time 0 seconds, when does the growth of each strain pass that of the others? 5 Graph the growth of each disease on your graphics calculator or by using the Graph option in the spreadsheet. 6 In your own words, describe in writing the shape of the exponential graph generated for each disease.
The index laws Working mathematically Reasoning and reflecting: Multiplying terms with the same base 1 Use a calculator to find the value of:
a c
i 23 × 24 i 44 × 42
ii 27 ii 46
b d
i 35 × 33 i 56 × 53
ii 38 ii 59
2 What do you notice about each pair of answers in Question 1? 3 Is it true that 25 × 27 = 212? Explain. 4 State whether each of the following are true (T) or false (F). Explain each choice. b 74 × 78 = 732 c 45 × 48 = 440 a 26 × 24 = 210 5 Copy and complete the following. … … a 47 × 43 = 4 b 53 × 54 = 5 … d 83 × 82 = … e k3 × k8 = k
d 37 × 312 = 319 …
c 68 × 65 = 6 f m3 × m7 = …
6 Use a calculator to find the value of:
a c
i 23 × 25 i 37 × 34
ii 48 ii 911
b d
i 54 × 56 i 62 × 63
ii 2510 ii 365
7 Use your results from Question 6 to decide whether these are true (T) or false (F): a 23 × 25 = 48 b 54 × 56 = 2510 c 37 × 34 = 911 d 62 × 63 = 365 8 Write true (T) or false (F) for each of the following. a 53 × 58 = 2511 b 27 × 210 = 217 3 10 30 d 4 ×4 =4 e 53 × 54 = 257
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c 73 × 72 = 75 f 33 × 39 = 312
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Law 1: Multiplying terms with the same base
SkillBuilder 11-02 The first index law
Consider 24 × 23 = (2 × 2 × 2 × 2) × (2 × 2 × 2) =2×2×2×2×2×2×2 = 27 4 + 3 But 2 = 27 4 3 ∴ 2 × 2 = 24 + 3 = 27
When multiplying terms with the same base, add the powers: am × an = am + n
Proof:
a×a×…×a×a×a×…×a m factors n factors
am × an =
a×a×…×a m + n factors = am + n =
Example 4 Simplify the following, expressing your answers in index form. a 63 × 67 b 5 × 53 c y4 × y8
Solution a 63 × 67 = 63 + 7 = 610
b 5 × 53 = 51 × 53 = 51 + 3 = 54
c
y4 × y8 = y4 + 8 = y12
Example 5 Simplify the following. a 3p4 × 2p6
b 5e2f × 3e4f 5
Solution a 3p4 × 2p6 = (3 × 2) × (p4 × p6) = 6p4 + 6 = 6p10
b 5e2f × 3e4f 5 = (5 × 3) × (e2 × e4) × (f × f 5) = 15e2 + 4f 1 + 5 = 15e6f 6
Exercise 7-02 Example 4
1 Simplify (giving answers in index notation): a 103 × 102 b 10 × 104 e 8 × 83 × 84 f 54 × 5 × 54 7 13 i 11 × 11 j 2 × 23
c × g 6 × 62 × 63 × 64 k 34 × 3 × 37
d ×7 h 44 × 44 × 44 l 72 × 75 × 7
2 Simplify: a x × x4 e p10 × p10
c w7 × w g y × y3 × y2
d b3 × b10 h m3 × m × m 4
b g4 × g4 f r×r
32
35
74
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CAS 7-02 Index multiplication
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Example 5
3 Simplify: a 3p2 × 2p5 d h3 × 5h8 g 5n8 × 6n8 j 10p3 × 5p2q m 5a3c2 × 2b4c
b e h k n
4y10 × 3y 2 3q3 × 8q8 2b3 × 15b6 8a3b2 × 2b3a 10p3q8 × qp2
c f i l o
6m × 3m8 2a2 × 5a5 3e4 × e6 4w 5y 2 × 5w 4y3 4g3h2 × 5gh4
4 Write true (T) or false (F) for each of the following. a 53 × 37 = 1510 b 72 × 82 = 564 c 3 × 72 = 212 d 43 × 47 = 410 2 4 2 5 4 9 2 3 15 7 e 3 × 2 × 3 × 2 = 3 × 2 f 5 × 5 = 25 g 2 × 28 = 215 3 5 8 2 3 6 h 7 × 7 = 49 i 4 × 3 = 12 j 54 × 32 × 37 × 5 = 39 × 55 5 Simplify and evaluate: a 23 × 25 b 23 × 52 3 3 e 3 ×3 f 53 × 23 SkillBuilder 11-06 Multiplying terms with indices
6 Simplify: a x 4 × x3 × x2 e 5qp × 4q2 × 5p3 i 32y × 3y
c 102 × 210 g 102 × 103
d 53 × 35 h 210 × 103
b y 6 × x3 × y c 5 × 3n × 4n2 f (a4 × b3) × (a4 × b 2) g 4a × 4b j (p + q)2 × (p + q)3 k (x – y) × (x – y)2
d 5 × m × 4n2 h 2x + 1 × 2x l (a + 3)n × (a + 3)
Working mathematically Reasoning and reflecting: Dividing terms with the same base 1 Use a calculator to find the value of:
a c
i 210 ÷ 27 i 37 ÷ 32
ii 23 ii 35
b d
i 55 ÷ 53 i 68 ÷ 64
ii 52 ii 64
2 What do you notice about each pair of answers? 3 Is it true that 38 ÷ 36 = 32? Explain. 4 State whether each of the following are true (T) or false (F). a 310 ÷ 36 = 34 b 48 ÷ 42 = 44 c 212 ÷ 23 = 24
SkillBuilder 11-03 Division of terms with indices
d 610 ÷ 65 = 65
5 Copy and complete the following. … … b 58 ÷ 56 = 5 a 27 ÷ 23 = 2 d 311 ÷ 36 = … e y8 ÷ y5 = …
c 67 ÷ 62 = 6 f m12 ÷ m10 = …
6 Write true (T) or false (F) for the following. b 10 6 ÷ 102 = 13 a 106 ÷ 102 = 104 12 3 9 d 8 ÷8 =8 e 410 ÷ 45 = 42
c 106 ÷ 102 = 103 f 66 ÷ 62 = 64
…
Law 2: Dividing terms with the same base Consider
57 57 ÷ 54 = ----54 1
1
1
1
1
1
5×5×5×5×5×5×5 = ---------------------------------------------------------15 × 5 × 5 × 5 1
=5×5×5 = 53
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57 − 4 = 53
But
57 ∴ 57 ÷ 54 = ----- = 57 − 4 = 53 54 When dividing terms with the same base, subtract the powers: am am ÷ an = ------ = am − n an
Proof: am am ÷ an = -----an 1 1 a×a×a×a×a×…×a = ------------------------------------------------------------a ×1 a × a × a × … × a 1 = a × a × … × a(m − n factors) = am − n
(m factors) (n factors)
Example 6 Simplify the following, expressing your answers in index form. a 45 ÷ 43
Solution a 45 ÷ 53 = 45 − 3 = 42
10 7 b -------10 4
c y12 ÷ y3
10 7 b -------- = 107 − 4 10 4 = 103
c y12 ÷ y3 = y12 − 3
b 15m8 ÷ 3m2
c 30a5b7 ÷ 10a2b5
= y9
Example 7 Simplify the following. a k7 ÷ k
Solution a k7 ÷ k = k7 ÷ k1 = k7 − 1 = k6
5
15m 8 b 15m8 ÷ 3m2 = ------------2 1 3m = 5m8 − 2 = 5m6
3
30a 5 b 7 c 30a5b7 ÷ 10a2b5 = ----------------2 5 1 10a b = 3a5 − 2b7 − 5 = 3a3b2
Just for the record Remember that taxi Indian mathematician Srinivasa Ramanujan (1888–1920) loved working with numbers. One day he was visited by a friend in a taxi numbered 1729. When Ramanujan heard the number, he immediately said ‘1729 is a very interesting number as it is the smallest number that can be expressed as the sum of two cubes in two different ways.’ This means that we can write 1729 = x3 + y3 Here is one of the possible ways: 1729 = 103 + 93 Find the other. I NDI CE S
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Exercise 7-03 Example 6
CAS 7-03 Index division
1 Simplify, giving your answers in index form: 58 a ----52 e 105 ÷ 105
9 12 b ------93 f 85 ÷ 8
680 ÷ 620
i
j
2 27 c ------23 g 2015 ÷ 205
815 ÷ 811
d 74 ÷ 73 2 20 h ------2 l 218 ÷ 211
k 312 ÷ 36
2 Simplify: h 20 a ------h4 e m16 ÷ m16 e30 ÷ e10
i Example 7
y8 b ----y2 f n7 ÷ n j
a 12 c ------a4 g t18 ÷ t 9
d9 ----d5
f
18g60 ÷ 6g4
i
12g 12 -------------6g 6
a6 b3 ----------a2 b2 n 36y8x7 ÷ 12x3y
4 Write true (T) or false (F) for the following. a 103 ÷ 22 = 51 b 84 ÷ 44 = 22 e 10 9 ÷ 103 = 106 5 Evaluate: a 210 ÷ 25 g 45 ÷ 210 SkillBuilder 11-04 Using the second index law
f
74 ÷ 72 = 12
b 45 ÷ 23
c 33 ÷ 23
h 203 ÷ 53
i
g 16h10 ÷ 8h
30x 4 d ----------x3 h 15y8 ÷ 15y4
36 p 8 q 3 k -----------------4 p4 q o 44e4f 10 ÷ 4ef 2
100 f 2 g 4 --------------------5 f g2 p 30k7m4 ÷ 6k6m2
c 1210 ÷ 1210 = 1
d 158 ÷ 154 = 152
20 4 g -------- ÷ 42 52
h 123 ÷ 33 = 41
c 24r 8 ÷ 3r 2
j
m 20m15n ÷ 2m14
10 3 d -------23
106 ÷ 54
j
49 ÷ 83
6 Simplify: x4 × x3 ---------------x2 d 4a ÷ 4b
y 10 b -------------3 y ×y e 2x + 1 ÷ 2x
× g -------------------------10m 6
× h -------------------------3n 3 × 4n 5
a
4m 3
5m 7
w 25 h -------w l w 24 ÷ w 6
k p15 ÷ p10
3 Simplify the following. a 10y15 ÷ 5y3 b 20w9 ÷ 4w3 10m 10 e --------------2m 2
d b16 ÷ b15
6n 16
54 ----54
e
12 5 k -------68 c f
8n 4
l
a5 × a3 ----------------a × a4 32y ÷ 3y
i
p6 30 p 4 --------- × -----------5p 6 p2
Working mathematically Reasoning and reflecting: Powers to powers 1 Use a calculator to find the value of:
a c
i (23)2 i (52)3
ii 26 ii 56
b d
i (34)3 i (25)4
2 What do you notice about each pair of answers in Question 1?
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ii 312 ii 220
f
2 10 ------52
l
310 ÷ 272
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3 Is it true that (27)3 = 221? Explain. 4 State whether each of the following is true (T) or false (F). a (35)3 = 315 b (23)2 = 25 2 5 10 d (4 ) = 4 e (33)6 = 318
c (210)4 = 214 f (52)4 = 56
5 Copy and complete: … a (37)2 = 3 … d (a3)4 = a
c (45)2 = 4 f (k 4)6 = …
…
…
b (52)6 = 5 e (83)7 = …
6 State whether the following are true (T) or false (F). a (25)7 = 212 b (28)3 = 224 3 7 7 d (7 ) = 7 e (84)5 = 89
c (53)4 = 512 f (66)5 = 630
Law 3: Raising a power to a power Consider (42)5 = 42 × 42 × 42 × 42 × 42 = (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4) × (4 × 4) =4×4×4×4×4×4×4×4×4×4 = 410 But 42 × 5 = 410 ∴ (42)5 = 42 × 5 = 410
When raising a term with a power to another power, multiply the powers: (am)n = am × n
Proof: (am)n = am × am × … × am n factors =a×a×…×a
×
m factors
a×a×…×a
×…×a×a×…×a
m factors
m factors
n lots of m factors
= am × n
Example 8 Simplify the following, expressing your answers in index form. a (23)5 b (y2)14
Solution a (23)5 = 23 × 5 = 215
b (y 2)14 = y 2 × 14 = y 28
Law 4: Powers of products and quotients
SkillBuilder 11-07 Multiplying expressions with brackets
Consider (2 × 3)5 = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) = 2 × 2 × 2 × 2 × 2 × 3 ×3 × 3 × 3 × 3 = 25 × 35 I NDI CE S
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4 5--- = 5--- × 5--- × 5--- × 5-- 3 3 3 3 3
Also
5×5×5×5 = -----------------------------3×3×3×3 4
5 = ----4 3
Powers of products and quotients: (ab)n = anbn
Proof:
and
a--- b
n
an = -----bn
ab × ab × … × ab n factors =a×b×a×b×…×a×b a×a×…×a×b×b×…×b = n factors n factors = an × bn
(ab)n =
a n a a a Also --- = --- × --- × … × -- b b b b
n factors a × a × … × a ( n factors ) = ------------------------------------------b × b × b × … × b ( n factors ) n
a = ----n b
Example 9 Simplify each of the following. a (2k)5
b (5m4)3
c
m ---- 4
3 4
3
2w d --------- 3
Solution a (2k)5 = 25 × k5 = 32k5
c
3 m3 m ---- = ----- 4 43
m3 = ----64
b (5m4)3 = 53 × (m4)3 = 125 × m4 × 3 = 125m12 2w 3 4 ( 2w 3 ) 4 d --------- = ------------- 3 34 24 × ( w3 )4 = ------------------------34 16w 12 = --------------81
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Just for the record The house fly The female common house fly, Musca domestica, can lay up to 1000 eggs at a time. In three weeks these reach maturity and are ready to breed. Huge populations would result if all the descendants of a single pair of house flies survived and reproduced. Fortunately, this is not the case as the mortality rate is very high. The few house flies we see are the true survivors. Over the 13 weeks of summer, how many descendants could a single pair of house flies produce, assuming that each pair (original and descendants) mates only once? (Give your answer in index form.)
Exercise 7-04 Example 8
1 Simplify, giving your answers in index form: a (43)2 b (52)8 c (33)4 4 3 0 2 f (9 ) g (10 ) h (64)5 1 5 3 0 k (3 ) l (7 ) m (22)10
d (27)4 i (53)5 n (132)2
e (21)2 j (25)10 o (44)4
2 Simplify each of the following. Give your answers in index form. a (e2)4 b (t 5)5 c (y3)7 d (c)5 4 4 0 6 6 3 f (y ) g (h ) h (p ) i (w4)1 k (n3)8 l (d 3)3 m (k5)10 n (d 3)4
e (m7)5 j (x 1)10 o (a8)8
3 Simplify the following. b (5m)2 a (2d)4 f (2w5)3 g (10d 5)4 4 5 k (3f ) l (2c3)10
e (5m6)5 j (6d 6)2 o (8w3)2
Worksheet 7-02 Indices puzzle
Example 9
c (4y5)2 h (3e7)3 m (3h5)4
d (3x2)4 i (2b4)1 n (6k)2
SkillBuilder 11-08 The fourth index law
4 Simplify each of the following. 5
a
--e- 2
e
f ---- 3
i
2k ------ 5
b
--x- 7
f j
2 4
3 2
2
c
3m ------- 2
n 5 8 -----2 p
g
3r 4 2 - ------ c2
k
3
2
d
5h ------ 6
w 2 5 -----3- t
h
am ------- c
a 2 b 4 ------- d5
l
5c 2 3 -------3- 3x
4
5 Simplify the following, giving your answers in index form. a (m3)10 b (5t)3 c (−2)8 e (y3)12 f (4w5)4 g (−2d)5 2 3 3 2 i (−3p ) j (−5m ) k (3f 5)5
d (−x)3 h (210)10 l (−m2)4
6 Evaluate: a (23)2 e (−2)3
d (−5)2 h (−52)3
i
3--- 2
2
b (−32)2 f (−42)3 j
2--- 5
2
c (102)3 g (−34)2 k
5--- 2
3
l
− 3--- 4
2
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7 Simplify each of the following. c
a 4 7 ----3- d
d
2 y 3 3 - ------ x2
g
p 2 q 3 5 ---------- t4
h (w2ak 4)8
j
(−2p2w3)4
k (a2d 3y5)0
l
n
m2 n3 5 - − ---------- 2 y5
o (4k3m)3
p (8k 4y5)2
a (l 3m5)6
b (x2y4)5
e (−we2k3)3
f
i
(−3m2n)5
m
d e f ------------- 4
2 5
3
q (3a3df 4)5
r (6d 5p2)4
8 Simplify: a (52)x d (b3)4 ÷ (b4)2
s
2 4
3a y 4 5 - − ---------- b2
t
5 3
h
a 2 c 4 0 - --------- d7
k p 3 2 --------4- 3q
c ((x + 1)2)3 f (6n4)2 × (3n2)2
b (5x)2 e (b3 )4 × (b4)2
g (6n4)2 ÷ (3n2)2
− m ------ n
(3 x ) ---------------2 4 (x )
5
i
5
5 y × (2 y) ---------------------------2 3 2 8y × (y )
Working mathematically Questioning and reasoning: The power of zero 1 Copy and complete the following sentence. A number remains unchanged when multiplied or divided by … 2 Copy and complete the following. a 34 × 30 = 3? b 52 × 50 = 5? 5 0 ? e 4 ×4 =4 f 50 × 57 = 5?
c 20 × 27 = 2? g 30 × 35 = ?
d 70 × 73 = 7? h 80 × 86 = ?
3 Copy and complete the following. a 25 ÷ 20 = 2? b 35 ÷ 30 = 3? 6 0 ? e 5 ÷5 =5 f 84 ÷ 80 = 8?
c 42 ÷ 40 = 4? g 157 ÷ 150 = ?
d 93 ÷ 90 = 9? h 68 ÷ 60 = ?
4 Copy and complete the following tables. Compare your answers with those of other students. a
Index Number form
b
Index Number form
c
Index Number form
25
32
35
24
16
34
84
23
33
83
22
32
82
21
31
81
20
30
80
243
85
5 Look at your results from Questions 1 to 4. Can you suggest a value for any number (or base) raised to the power of zero (for example, 30 = ?, 50 = ?)? Explain.
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The zero index
SkillBuilder 11-05 Raising to the power of 0
3
5 Consider 53 ÷ 53 = ----3 5 1
1
1
5×5×5 = --------------------1 5 × 51 × 51 =1 53 ÷ 53 = 53 − 3 = 50 ∴ 50 = 1
But
Any number raised to the power of zero is equal to 1: a0 = 1
Proof: 1
1
1
1
a×a×a×…×a am ÷ am = ------------------------------------------1 a ×1 a ×1 a × … × a1 but
am
(m factors) (m factors)
=1 ÷ = am − m = a0 ∴ a0 = 1 am
Example 10 Simplify the following. a 70
Solution a 70 = 1
b (−3)0
c m0
b (−3)0 = 1
c m0 = 1
Example 11 Simplify: a (ab)0
Solution
a (ab)0 = 1
b (5k)0 b (5k)0 = 1
Example 12 Simplify: a 5d 0
Solution
a 5d 0 = 5 × d 0 =5×1 =5
b (3y)0 + 3y0 b (3y)0 + 3y0 = 1 + 3 × y0 =1+3×1 =4
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Exercise 7-05 Example 10
1 Simplify the following. a 80 b (−2)0 2 0 e --- 3 i (−14)0
Example 11
2 Simplify: a (km)0 p e --- q
Example 12
(−6)0 0 5--- 4
f j
3--- 4
f
d m0
g (−700)0
h (1 000 000)0 1 0 l − --- 2
k a0
b (x2y)0
0
c d0
0
3 Simplify the following. a 70 + 20 b 3y0 e (5t 2)0 f (6x)0 + 20 0 i 12u ÷ 3 j 32 × 50
c (xyw)0
d (−ab)0
g (7y)0
h (9cd)0
c −(4m)0 g 2m0 + (2m)0 k (5a)0 + 4
d 3 × (5d)0 h 2w0 × 3p0 l 8b0 − (3b)0 1 0 1 p --- + --- y 0 2 2 t (3x3)3 ÷ x9 x 7 × 4k 0
m 6h0 − (6h)0
n −7c + 4c0
o (3e2)0 − (10e)0
q 1000 − 10000 u 60m5n3 ÷ 12mn3
r 3f 0 + 4 − (5f)0 v 12p0 ÷ (2p)0
s 36q5 ÷ 12q 5 w (a2b3)0
Working mathematically SkillBuilder 11-09 Exercising the four index laws
Applying strategies and reasoning: Negative powers 1 a Copy and complete the following table of descending powers of 10. Use your calculator if necessary. (Don’t be alarmed if the calculator gives decimal answers.) What rule did you use to complete the pattern? Powers to ten
106
…
Decimal form
1 000 000
…
100
10 −1
10 −2
…
0.1
0.01
…
10 −6
b To see the hidden pattern clearly, you will need to change the decimals into fractions. Copy and complete the following table. Express each decimal as a fraction, then write it as a power of 10. (The first two have been done for you.) Powers of ten
10 −1
10 −2
…
Decimal form
0.1
0.01
…
Fraction form
1 -----10
1 --------100
…
Fraction form with powers of ten
1 -------1 10
1 -------2 10
…
10 −6
c Look carefully at the fractions written as powers of 10. What do you notice when you compare them with the corresponding negative powers of 10? Write down your findings. Write 10 −7 and 10 −8 as fractions using the power of 10. d What does this tell you about negative powers? e Write down what you have learnt about raising a number to a negative power.
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2 a Copy and complete the table below. Use your calculator to express each power of 5 as whole numbers or a fraction. 53
52
51
50
5−1
5−2
5−3
5− 4
5−5
1 1 ------ = ----25 52
125
1 1 b 5−2 can be written as ------ , or as ----- . Use the table to write each of the following in two 2 25 5 ways. i 5−3 ii 5− 4 iii 5−5 c Write each of the following in two ways. ii 7−3 iii 2− 6 i 4−2 10 4 10 × 10 × 10 × 10 3 Consider: -------- = ------------------------------------------------------------------------------10 7 10 × 10 × 10 × 10 × 10 × 10 × 10 1 = -----------------------------10 × 10 × 10 1 = -------10 3 10 4 -------- = 104 ÷ 107 But 10 7 (Using Law 2) = 10 −3 1 So 10 −3 = -------10 3 Using this method, simplify (in the two ways): 1 23 a ----- to show that 2−5 = ----28 25
34 1 b ----- to show that 3−1 = --3 35
2
a4 1 d ----- to show that a −2 = ----6 a a2
5 1 c ----- to show that 5− 6 = ----8 5 56
1 4 The reciprocal of 35 is ----- = 3−5. 35 Use negative indices to write the reciprocals of the following. a 24 b 52 c 4 d k5 Compare your answers with those of other students.
e m3
The negative index Consider
24
÷
27
SkillBuilder 11-13 Division with a larger index in the denominator
24 = ----27 1
1
1
1
2×2×2×2 = ---------------------------------------------------------1 2 ×1 2 ×1 2 ×1 2 × 2 × 2 × 2 1 = --------------------2×2×2 1 = ----23
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24 ÷ 27 = 24 − 7 = 2−3 1 ∴ 2−3 = ----23
But
A negative power or index gives a fraction (with numerator 1): 1 a −m = -----am
Proof:
a0 a0 ÷ an = ----an 1 = ----- (since a0 = 1) an a0 ÷ an = a0 − n = a −n 1 ∴ a −n = ----an
But
Example 13 Express using positive indices: a 3−1 b 4−3
c k −5
Solution 1 a 3−1 = ----31 1 = --3
b
1 4−3 = ----43
1 c k −5 = ----k5
Example 14 Express using positive indices: a 3k −5 b a2b−3
c (5m)−2
Solution a 3k −5 = 3 × k −5 3 1 = --- × ----1 k5 3 = ----k5
b a2b −3 = a2 × b −3 3 (3 = --- ) 1
a2 1 = ----- × ----1 b3 a2 = ----b3
Example 15 Evaluate 2 −3, leaving your answer as a fraction.
Solution 1 2 −3 = ----23 1 = --------------------2×2×2 1 = --8
236
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1 c (5m)−2 = --------------( 5m ) 2 1 = ------------25m 2
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Negative powers of quotients 2--- 3
Consider
-1
1 = ---------2 --3 2 = 1 ÷ --3 3 = 1 × --2 3 = --2
4--- 5
and
-2
1 = ----------2 4--- 5 1 = ---------16 -----25 16 = 1 ÷ -----25 25 = 1 × -----16 52 = ----42 5 2 = --- 4
a--- b
-1
b = --a
and
a--- b
-n
b n = --- a
Proof: a--- b
-1
1 = ----------a --b a = 1 ÷ --b b = 1 × --a b = --a
and
a--- b
-n
1 = ---------- a--- n b an = 1 ÷ ----bn bn = 1 × ----an bn = ----an b n = --- a
Example 16 Simplify the following and evaluate if possible. 4 -1 a --- 5
3 -2 b --- 5
c
-3 2a ------ b2
3 -2 5 2 b --- = --- 5 3
c
-3 b2 3 2a ------ = ------ b2 2a
Solution 4 -1 5 a --- = -- 5 4 = 1 1--4
25 = -----9 = 2 7--9
( b2 )3 = ------------( 2a ) 3 b6 = -------8a 3
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Exercise 7-06 Example 13
CAS 7-04 Negative indices Example 14
Example 15
SkillBuilder 11-14 The fifth index law
1 Express using positive indices: a 5−2 b 3−7 − 4 e 10 f m −1 − 4 i 20 j (−11)−1 2 Express using positive indices: a 4d −1 b 3x −5 2 − 4 f mn g wy −2 2 − 3 k 12y m l a − 4m2
d 8−2 h w −2 l c −6
c 2d −3 h 4ac −1 m d −3y3
d 4m −2 i 3p −2 n 4xy −3
e ab −2 j 15kw − 4 o v −1m −2
3 Write each of the following using positive indices. a (2m) −1 b (xy) −1 c (4h) −2 − 2 − 3 e (3h) f (4k) g (2c) − 4
d (5k) −3 h (8y) −1
4 Evaluate the following, leaving your answers as fractions. a 3 −2 b 4 −3 c 6 −1 − 1 − 5 e 11 f 2 g 4 −2
d 7 −2 h 10 −2
5 Express using negative indices: 1 1 a ---b ---m w 1 1 f ----g ----n4 34 2 4 k --l ---a t2 1 p -----7e
Example 16
c 4−1 g h −3 k k −8
1 q -------3a 2
1 c --8
1 d --9 1 i ----e3 5 n --d
1 h --------10 -3 2 m -----w5 r
5 ---------3m 4
1 e ----22 1 j ---t2 1 o -----2y
1 --------8 p3
s
t
2 -------3k 6
6 Evaluate: 1 -1 a --- 3 e
2--- 3
1 -2 b --- 4
-1
f
3--- 4
-1
c
2--- 3
-2
2 -3 d --- 5
1 -5 g ------ 10
5 -1 h --- 4
7 Simplify the following and evaluate if possible. 4 -1 a ---- w
m -1 b ---- n
-1
e
--k- 3
i
2d ------ 5
-2
f
--x- 3
j
h2 5 − ------3 m
-2
c
1--- 4
a2 g ----- 4
N E W C E N T U R Y M A T H S 9 : S T A G ES 5.2/ 5.3
4 -1 d --- 5
-3
a2 c3 k ---------- 4
8 Simplify each of the following, using positive indices. a y5 × y −2 b e −3 × e7 c m × m −1 f 5a −2 × 6a3 g 5x − 2 × 2x h 30e −3 × 2e −1 2 − 1 − 2 − 1 k 2r ÷ 8r l 2t ÷ 8t m (h −1) 4
238
-1
4 -2 h − --- 3 -3
d n6 × n −5 i 8p −1 ÷ 2p2 n (b)−3
l
5d 2 - ------- p3
-3
e 4g3 × 3g −1 j 8q ÷ 2q −2 o (5x −1) 2
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9 Simplify the following and express your answers in positive index form. a x3y4 × x −3y −5 b p−4q −1 × 5p2q−3 c (m2n3)−2 3 5 5 3 2 3 − 5 − 1 d w p ÷w p e m n ÷m n f 4a3bc2 × −2a −5b −3c−2 3 2 7 4 − 2 − 3 g 8xy ÷ 4x y h (6m ) × 9m i p2q × p −3q −1 ÷ p4q3 3 − 1 2 3 2 2 − 3 − 1 2 − 2 j 8a h ÷ −4ah ÷ a h k (a k ) × (a k ) l 4x −3y −1 ÷ 8xy3 × 5x −1 4 − 3 − 5 4 − 2 − 1 − 3 7 m 4r t × 5r t n -15ab ÷ 5a b ÷ −6ab o (d −3h −1)−1 ÷ −4d 3h2 p (2v 3w −2)5 ÷ 8v 2w −7 q 81a−3e −4 ÷ (3a 2e −1)4 r (c −1d −3)3 ÷ (c −1d 2)−4 10 Evaluate the following, leaving your answers as fractions. a 23 × 2 −4 b (32)−3 ÷ (3−3)3 − 2 − 1 d 3 ÷2 ×6 e (4−2)2 ÷ (2−2)3
c 5−1 ÷ 2−1 f 32 × (3−2)2
SkillTest 7-01 Squaring a number ending in 5, 1 or 9
Skillbank 7 Squaring a number ending in 5, 1 or 9 Squaring a number ending in 5 The square of a number ending in 5 always ends in 25. For example, 352 = 325, and 1052 = 11 025. A simple calculation trick requires three steps: Step 1: Delete the 5 from the number. Step 2: Multiply the remaining number by the next consecutive number. Step 3: Write ‘25’ at the end of the product. 1 Examine these examples: a 352 Deleting the 5 from 35 leaves just 3. Multiplying 3 by the next consecutive number: 3 × 4 = 12 Writing ‘25’ at the end: 1225 352 = 1225 b 1052 Deleting the 5 from 105 leaves 10. 10 × 11 = 110 11 025 1052 = 11 025 2 Now calculate these: a 252 e 1152 i 1.52
b 552 f 7.52 j 652
c 452 g 952 k 1552
d 852 h 1952 l 2452
Squaring a number ending in 1 The square of a number ending in 1 always ends in 1. For example, 412 = 1681, and 712 = 5041. A simple calculation trick requires three steps: Step 1: Subtract 1 to round down to the nearest 10 and make a new number. Step 2: Square the new number. Step 3: Add the new number and the next consecutive number to the square.
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3 Examine these examples: a 412 Round down to 40. Squaring 40: 40 2 = 1600 Adding 40 and 41 to 1600: 1600 + 40 + 41 = 1681 412 = 1681 b 712 70 2 = 4900 4900 + 70 + 71 = 5041 712 = 5041 4 Now calculate these: a 212 b 1012 2 e 5.1 f 612
c 312 g 2012
d 912 h 1.12
Squaring a number ending in 9 The square of a number ending in 9 also ends in 1. For example, 292 = 841, and 992 = 9801. A simple calculation trick requires three steps: Step 1: Add 1 to round up to the nearest 10 and make a new number. Step 2: Square the new number. Step 3: Subtract the new number and the previous consecutive number from the square. 5 Examine these examples: a 292 Rounding up gives 30. Squaring 30: 30 2 = 900 Subtracting 30 and 29 from 900: 900 − 30 − 29 = 841 292 = 841 b 992 100 2 = 10 000 10 000 − 100 − 99 = 9801 992 = 9801 6 Now calculate these: a 592 e 1092
b 692 f 4.92
c 892 g 792
d 192 h 11.92
Note: By combining and adapting the methods for squaring numbers ending in 5, 1 and 9, it is also possible to square a number ending in 4 or 6. Bonus trick: Squaring a two-digit number beginning with 1 This calculation trick requires three steps: Step 1: Double the units digit and add 10. Step 2: Multiply by 10. Step 3: Add the square of the units digit. 7 Examine these examples: a 172 Doubling the units digit and adding 10: 2 × 7 + 10 = 24 Multiplying by 10: 24 × 10 = 240 Adding the square of the units digit: 240 + 72 = 240 + 49 = 289 172 = 289
240
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b 142 2 × 4 + 10 = 18 18 × 10 = 180 180 + 42 = 180 + 16 = 196 142 = 196 8 Now calculate these: a 122 d 192
b 132 e 112
c 182 f 1.62
Working mathematically Reasoning and reflecting: Fractions as powers (Spreadsheet optional) 1 Copy and complete this table of square numbers and their square roots. Square number
Square root
1
1
4
2
9
3
100
10 1 ---
2 Use your calculator to evaluate 25 2. (25 Now evaluate: a
1 --36 2
b
(
^
1 --64 2
c
--ab
1
c
)
2
1 --81 2
=
)
1 --100 2
d
3 Look at your calculator answers. Compare them with your answers to Question 1. 1 ---
1 ---
Write down what you notice. Predict the values of 12 2 and 144 2 . 1 4 What have you learnt about the fractional power --- ? 2 5 Repeat this investigation for fractional cube numbers and their cube roots. Copy and complete this table. Cube number
Cube root
1
1
8
2
27
3
1000
10 1 ---
6 Use your calculator to evaluate 8 3 .
(8
^
(
1
ab --c
3
)
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)
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Now evaluate: 1 ---
1 ---
a 27 3
1 ---
b 64 3
1 ---
c 512 3
d 1000 3
7 What do you notice about your answers to Questions 5 and 6? 1 8 Explain what the fractional power --- means. 3
Using technology Calculating square roots
Spreadsheet 7-02 Calculating square roots
For computer spreadsheets Step 1: Set up your spreadsheet as shown. C
D
E
5
Number
Square root of a number
Number to power of 1/2
6
1
=SQRT(C6)
=C6^(1/2)
7 8
=C6+1
34
Step 2: Copy the formulas in cells C7, D6 and E6 to row 34. Step 3: Print your spreadsheet and paste it in your workbook. 1 Use your spreadsheet to find: a b 9 13
c
25
d
29
2 Use your spreadsheet to find: a
1 --92
b
1 --13 2
c
1 --25 2
d
1 --29 2
3 a Compare your answers to Questions 1 and 2. Write what you notice. 1 ---
1 ---
b Predict the values of 36 2 and 64 2. 4 • • •
In cell F5, insert the heading Number to the power 1/3. In cell F6, enter the formula =C6^(1/3). Copy cell F6 down to row 34.
5 Suggest a meaning for: a
SkillBuilder 11-15 Simplifying fractions
1 --n2
b
1 --n3
The fractional index The square root ( 64 = 8 because
82
) of a number is the value that, when squared, gives the number. For example, = 8 × 8 = 64.
The cube root ( 3 ) of a number is the value that, when cubed, gives the number. For example, 3
125 = 5 because 53 = 5 × 5 × 5 = 125.
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Consider
But
1 --42
×
1+1 --- --42 2
1 --42
= = 41 =4 4× 4=2×2 =4 1 ---
∴ 42 = 4
1 Any number raised to the power of --- is the square root of that number: 2 1 ---
a2 = a
Proof: 1 ---
1+1 --- --2
1 ---
a2 × a2 = a2 = a1 =a a × a =a
But
1 ---
∴ a2 = a 1 ---
1 ---
1 ---
1 1 1 --- + --- + --3 3
We also have that 8 3 × 8 3 × 8 3 = 8 3 = 81 =8 3
But
8 ×3 8 ×3 8 =2×2×2 =8
(3 8 = 2 because 2 × 2 × 2 = 8)
1 ---
∴ 83 = 3 8
1 Any number raised to the power of --- is the cube root of that number: 3 1 ---
a3 = 3 a
Proof:
But
1 --a3
3
×
1 --a3
×
1 --a3
a ×3 a ×3
1 1 1 --- + --- + --3 3 3
=a = a1 =a a =a 1 ---
∴ a3 = 3 a
Example 17 Write each of the following with a fractional index. a 5 b 3 11
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Solution 1 ---
5 = 52
a
1 ---
b
3
11 = 11 3
b
3
k
b
3
k = k3
Example 18 Write with a fractional index: a g
Solution 1 ---
g = g2
a
1 ---
Example 19 Evaluate: 1 --400 2
a
b
1 --125 3
Solution 1 ---
1 ---
a 400 2 = 400 = 20 (because 202 = 400) (Calculator steps:
– √
b 125 3 = 3 125 = 5 (because 53 = 125)
=
400
)
(Calculator steps:
3
– √ 125
Exercise 7-07 Example 17
CAS 7-05 Fractional indices Example 18
1 Write the following, using fractional indices. e
35
b
20
f
3
512
c
3
10
d
g
3
100
h
c
3
8k
d
3
15 72
2 Write the following, using fractional indices. a e
Example 19
8
a
m 3
b
9y
f
3
w xy
g
18 f
h
ab 3
10mn
3 Evaluate the following. 1 ---
1 ---
a 64 2
1 ---
e 0.04 2 (−8) 3
c 1000 3 1 ---
1 ---
f
0.125 3
j
(−729) 3
1 ---
i
1 ---
b 343 3
g (−64) 3
1 ---
1 ---
900 2 1 ---
c d2 1 ---
f
l
or 3 .
1 ---
1 ---
1 ---
k 196 2
b 83
e ( 4 p )3
1 ---
h 1024 2
1 ---
1 ---
4 Write the following, using either a 37 2
1 ---
d 625 2
( 100h ) 3
d 20 2 1 ---
g ( 3c 7 ) 2
1 ---
h w3
5 Evaluate, correct to 2 decimal places: a
244
1 --43
b
1 --82
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c
3
100
d
1000
=
)
07_NC_Maths_9_Stages_5.2/5.3 Page 245 Friday, February 6, 2004 2:17 PM
e
3
- 50
1 --1111 2
f
g
h ( - 0.008 )
3.6
1 --3
6 Simplify the following. a e
i
1 --n2
×
1 --n2
b
2 --3
5g × 2g
1 --3
1 --3 6 3
f
1 --4
× f
1 --4
3 --5
f
2h × 7g
j
40k 3 ÷ 8k 3
1 ---
( 8m n )
c x× 3 --5
g 1 ---
3 --5
1 --x3
d 2m × 3m
1 --2 2 4 (p q )
h
1 --2 6 6 ( 9a b )
l
27a 3 ÷ 3a
1 ---
k 25a ÷ 5a 2
2 --5
2 ---
7 Simplify: 3 ------
2 ---
a ( m 20 ) 10 3 ---
e
3 ---
b ( b 12 ) 3
c 2 ---
( 25q 10 ) 2
f
( 32h 10 ) 5
4 ---
( e8 ) 4
d ( p 5 q 15 ) 5 2 ---
g ( 8m 3 n 12 ) 3
5 ---
h ( 16x 8 y 4 ) 4
Just for the record Calculating square roots Before the advent of calculators and computers, square roots were often calculated manually in the classroom. Follow the working given below for
43 469
2 5 − 4 1 − 1
3 9 71 21
71 29 42 21 − 42 21 0
57 121.
ROOT (Answer line) Step 1: Group the digits of 57 121 in pairs from the right. Step 2: Find the largest square (4) that is less than 5. Subtract it from 5 and write its square root (2) on the answer line above the 5. Step 3: Bring down the next pair of digits, the 71. Step 4: Double the number (2) you have in the answer line, to give 4. Write the 4 on the left of the 171. Now, trying 41 × 1, 42 × 2, 43 × 3, 44 × 4, …, find the largest product that is equal to or less than 171. This turns out to be 43 × 3 = 129. Write in 129 and subtract it from 171, leaving 42. Step 5: Write the 3 in the answer line, above the 71. Step 6: Bring down the next pair of digits, the 21. Step 7: Double the number (23) you now have in the answer line, to give 46. Write the 46 on the left of the 4221. Now, by trying 461 × 1, 462 × 2, 463 × 3, …, find the largest product that is equal to or less than 4221. This turns out to be 469 × 9 = 4221. Write in the 4221 and subtract it from 4221, leaving a zero remainder. Step 8: Write the 9 in the answer line, above the 21.
∴ 57 121 = 239 Use this method to find
68 121 and 173 056 . Check your answers using a calculator.
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Using technology An iterative process to find the square root In the past, before calculators were developed, mathematicians used a process called iteration to give an approximate answer for the square root. One formula for finding an approximation for
M is 1 M xn + 1 = --- x n + ----- , where x0 is a first guess. 2 x n
Example Find a good approximation to Solution Guess:
500 , correct to two decimal places.
x0 = 20
1 500 Iteration 1: x1 = --- 20 + --------- = 22.5 2 20 1 500 Iteration 2: x2 = --- 22.5 + ---------- = 22.361 111 11 2 22.5 1 500 Iteration 3: x3 = --- 22.36 + ------------- = 22.360 679 78 2 22.36 This process can continue forever, increasing your accuracy each time. Hence
500 = 22.36, correct to two decimal places.
The second decimal place does not change for any further iterations. 1 Plan and trial an efficient keystroke sequence to enable you to do this iteration on your calculator. Write down your preferred sequence. 300 .
2 a Guess
b Evaluate
300 to three decimal places, using the iterative formula.
c Evaluate
300 on your calculator to check your answer.
71 .
3 a Guess
Spreadsheet 7-03 Square roots (iteration)
b Evaluate
71 to four decimal places, using the iterative formula.
c Evaluate
71 on your calculator to check your answer.
4 Design and trial a spreadsheet to perform this iterative process.
------ and m The fractional indices 1 n n 1 1 ---
---
Consider ( 2 5 ) 5 = ( 2 × 2 × 2 × 2 × 2 ) 5 1 ---
= 32 5 1 ---
and
246
1 5 × --5
( 25 ) 5 = 2 = 21 =2
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1 ---
∴ 32 5 = 2 = 1 ---
∴ 32 5 =
5
5
(since 2 × 2 × 2 × 2 × 2 = 32)
32
( 5 32 is read as ‘the fifth root of 32’)
32
Also
3 --5 ( 2 )5
and
( 25 ) 5 = ( 25 ) 5
3 ---
= 32 5
3 ---
1 --- × 3
( 25 )
or
1 3 × --5
1 ---
3
1 -- 5 5 = ( 2 )
= ( 32 3 ) 5 =
1 3 --- 5
= 32 =
5
32 ×
= ( 5 32 )
32 ×
5
1 ---
a
n
3 ---
3
a
and
1--n- m = a
m ---n
= (n a)
5
5
32 3
32
3
∴ 32 5 = ( 5 32 ) =
an =
1 ---
= ( ( 25 )3 ) 5
5
32
3
1 ---
m ----
a n = ( am ) n
or
m
=
n
am
Example 20 Write each of the following with a fractional index. a
7
m
b
4
a3
b
4
a = (a )
c
6
d 12
c
6
12
Solution a
7
m =m
1 --7
1 --3 4
3
=a
3 --4
d
1 --12 6
= (d ) = d2
Example 21 Evaluate each of the following. 1 ---
2 ---
a 64 3
5 ---
b 83
c
27 3
d 8
2 - --3
Solution a
1 --64 3
= 64 =4 3
b
2 --83
(3
= 8) = 22 =4
2
or
2 --83
=
3
8
2
= 3 64 =4 I NDI CE S
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5 ---
27 3 = ( 3 27 ) = 35 = 243
c
5
d
8
2 - --3
1 = ----2 ---
83
1 = -------------2 3 ( 8) 1 = ----2 2 1 = --4
Example 22 Evaluate the following using a calculator (correct to two decimal places). 1 --20 4
a
b
3 --54
Solution 1 ---
a 20 4 = 2.11 3 ---
b 5 4 = 3.34
x
– √ 20
(
3
SHIFT
(4
^
(5
= --ab
=
)
4
c
(
^
or 20
--ab
1
4
c
)
Example 23 Simplify: 1 --7 ( 16r ) 4
a
b
2 --6 ( 27k ) 3
b
2 --( 27k 6 ) 3
Solution 1 --( 16r 7 ) 4
a
=
1 1 --- 7 × --16 4 r 4 7 ---
2 2 --- 6 × --3 3
= 27 k = 9k4
= 2r 4
Exercise 7-08 Example 20
CAS 7-05 Fractional indices Example 21
1 Write each of the following with a fractional index. a
5
k
e
3
n
b 5
f
4
d e
5
c
3
g
6
y
18
d
1 ------4 n
4
x
16
1 h ---------3 7 a
2 Evaluate each of the following. 3 ---
1 ---
a 42
b 64 6 4 ---
f
248
64 3
3 ---
g 36 2
5 ---
c
2 ---
83
3 ---
h 81 4
N E W C E N T U R Y M A T H S 9 : S T A G ES 5.2/ 5.3
5 ---
d 27 3
e
100 x
j
8
2 ---
i
1000 3
1 - --3
=
)
07_NC_Maths_9_Stages_5.2/5.3 Page 249 Friday, February 6, 2004 2:17 PM
k 81
1 - --4
l
64
4 - --3
m 400
3 - --2
n 256
3 - --4
o 3125
4 - --5
3 Evaluate each of the following, correct to two decimal places. a
1 --15 4
e
3 - --100 4
b
1 --85
f
3 --16 5
c
5 --50 4
g
3 - --12 2
Example 22
d
2 - --63
h
2 --95
4 Simplify:
Example 23
a ( 16 p 4 ) e ( 8d 6 )
1 --4
b ( 8m 11 )
5 --3
1 --3
( 64n 12 )
f
2 ---
c ( 32d ) 5
4 --3
g ( 25q 10 )
d ( 27m 10 ) 3 --2
2 --3
h ( 1000e 3d 6 )
2 --3
Summary of index laws and properties am × an = am + n am ÷ an = am − n (am)n = am × n, ( ab ) n = anbn, a0 = 1
n n a--- = a---- b bn
1 a -1 b a -n b n a −n = ----- , --- = --- , --- = --- a a b a n b 1 --a2
= a,
m ----
1 --a3
a n = (n a)
m
=
3
=
n
1 --n
a, a = a
n
a
m
Exercise 7-09 1 Simplify the following, leaving your answers in index form. a 24 × 27 b 159 × 152 c 410 ÷ 47 8 7 8 5 e 9 ÷9 f 3 ÷3 ÷3 g (43)2 i (73)5 × 74 j 220 ÷ (23)4 k 710 ÷ (73)3
d 75 × 74 × 73 h (82)4 l (145)3 ÷ (142)3
2 Simplify: a x3 × x 4 e (m2)4 i t 7 × (t 2)3 m 4d 7 × 5d 6
d h l p
b f j n
w 8 × w9 (y 4)6 (d 4)4 ÷ d 12 30c12 ÷ 5c8
c g k o
m7 ÷ m2 a3 × a7 ÷ a8 q6 ÷ q4 × q5 24e8 ÷ 6e5 × 2e3
b
1 --16 2
c 50
k3 ÷ k (p2)3 ÷ p5 2b2 × 3b5 15m8 ÷ 5m3 ÷ 3m4
Worksheet 7-03 Indices squaresaw SkillBuilder 11-20–11-24 Review of index laws
3 Evaluate the following. a
30
d 7 × 20
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e
1 --27 3
+
40
f
50
+
4 Simplify the following. a (3m5)2 d p3q5 × pq 6 g 4w 5m × 5w3m4 j 4v 2 ÷ 2v
g
(72)0
h
a2w2 × a3w7 m5n2 × m2n3 2 × 3ab3 × 4a2b3 m7n3 ÷ m5n2
b e h k
c f i l
15x 5 y 7 n ----------------5x 2 y 4
m 24l 5d 3 ÷ 6l 2d 5 Evaluate: a 74 ÷ 73 e 52 ÷ 50 i (−2)3
100
b 85 ÷ 82 ÷ 8 f 33 + 22 j (−2)2
1 --83
+
1 --42
x3y × x2y4 2c2d 5 × 3c3d 3 (2y3)4 × 3y5 a13c6 ÷ a5c2
o 36g3h4 ÷ 9g2h2 × 2gh4 c 43 × 47 ÷ 48 g (33)2 ÷ 35 k (−5)0 + (−2)0
d 243 ÷ 243 h 87 × 84 ÷ 811 l 3 + 30
3y
d
3
10
1000
h
3
3m n
6 Express with a fractional index: a e
3
5
b
d
c
4p
f
xy
g
3
7 Express each of these using positive indices: a 7−8 b 2−10 c 15−1 f 9 −2 g 10 −3 h (ab)−1 − 1 3 − 5 k 10d l pq m mw −3
d y −3 i 4y − 8 n c2e−3
8 Evaluate the following, leaving your answers as fractions. a 2 −2 b 3 −1 c 20 −1 e 8 −3 f 10 − 4 g 6−4
f j
1 d ----74 1 h ----33 d 4−3 × 43
1 --3 2
g (9 ) k 35 ÷ 37
(43)4 (5 3)0
h 58 ÷ 53 l 63 ÷ 64
11 Simplify each of the following. a (4a3)2
e
4 -1 ---- 5a
i
2 25 ------ d 2
b (m2n3)5 2 ---
250
2k -------- m
10 -2 g ------- 7m
f
( 8c 3 ) 3
j
a 3 b 9 3 - --------- c6
5 ---
h ( 25w 5 ) 2
2 ---
3 ---
m (4k3n2)4
4y 5 3 d ---------- 3m 2
5 4
c
e (5x)−1 j (3a2) −1 o 8t 3m − 4 d 5 −3 h 3 −5
9 Rewrite these fractions, using negative indices: 1 1 1 a ----b --c -------3 2 10 3 4 1 1 1 e ---f ----g ----w d7 k4 1 10 Evaluate: --b (80)2 c ( 82 ) 3 a 2 −3 e 124 ÷ 124 i m0 + k 0
1 -2 n -------- 3g 2
N E W C E N T U R Y M A T H S 9 : S T A G E S 5.2/ 5.3
k
5
2
32m 10 5 ---
o ( 16x 8 ) 4
l
64 ------ y3
2 - --3
2a 3 4 p -------- c2
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Working mathematically Reasoning and reflecting: Repeated roots 16
This is the square root of the square root of 16.
16 = ( 16 )
1 1 --- --2 2
1 ---
= ( 16 ) 4
(the fourth root of 16)
=2 64 This is the cube root of the square root of 8.
3
1--2- 1--364 = ( 64 )
3
= ( 64 )
1 --6
(the sixth root of 64) (the sixth root of 64)
=2 1 Evaluate the following. a e i
4
81
b
256
f
256
j
4
625
c
256
g
5 3
1024
d
1 000 000
h
5 3
1024 1 000 000
Worksheet 7-04 Power calculations
6561 Worksheet 7-05 Binary number system
2 Rewrite each of the parts in Question 1 using a single fractional power.
Scientific notation What is scientific notation? Scientific (or standard) notation is a way of expressing very large or very small numbers using powers of 10. Its use originated in the early twentieth century, when scientists needed to describe very large values, such as astronomical distances, and very small values, such as the masses of atoms. It has the form m × 10 n, where m is a number between 1 and 10, and n is an integer.
Numbers written in scientific (or standard) notation are expressed in the form m × 10 n where m is a number between 1 and 10 and n is an integer.
SkillBuilder 11-16 Scientific notation SkillBuilder 11-17 Scientific notation for small numbers
Example 24 Express in scientific notation: a 2 700 000 000
b 0.004 67
c 5.78
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Solution
a 2 700 000 000 = 2 700 000 000 = 2.7 × 109 The power is 9 because the decimal point must move 9 places to the left to put the number in scientific notation. b 0.004 67 = 0.004 67 = 4.67 × 10 −3 The power is −3 because the decimal point must move 3 places to the right to put the number in scientific notation. c 5.78 = 5.78 × 100 The power is 0 because the decimal point does not need to move. Note: Large numbers are written with positive powers of 10, while small numbers are written with negative powers of 10.
Example 25 Express in decimal form: a 2.7 × 10 4
b 3.56 × 10 −2
Solution
a 2.7 × 10 4 = 2.7000 = 27 000 Since the power is 4, the decimal point moves 4 places to the right to convert to decimal form. b 3.56 × 10 −2 = 0.0356 = 0.0356 Since the power is −2, the decimal point moves 2 places to the left to convert to decimal form.
Exercise 7-10 Example 24
Worksheet 7-06 Scientific notation
1 Express each of the following numbers in scientific notation. a The distance from Earth to the sun is 152 000 000 km. b The world’s largest mammal is the blue whale, which can weigh up to 130 000 kg. c The diameter of an oxygen molecule is 0.000 000 29 cm. d The thickness of a human hair is 0.000 08 m. e Light travels at a speed of 300 000 000 m/s. f The nearest star to Earth, excluding the Sun, is Alpha Centauri, which is 40 000 000 000 000 km away. g The mass of a proton is 0.000 000 000 000 000 000 000 002 g. h The thickness of a typical piece of paper is 0.000 12 m. i The small intestine of an adult is approximately 610 cm long. j The diameter of a hydrogen atom is 0.000 000 0001 m. k The diameter of our galaxy, the Milky Way, is 770 000 000 000 000 000 000 m. l A microsecond means 0.000 001 s. m The Andromeda Galaxy is the most remote body visible to the naked eye, at a distance of 2 200 000 light years.
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2 Express in scientific notation: a 2400 b 786 000 e 7.8 f 348 000 000 i 300 000 000 j 80 m 456.3 n 8.007
c g k o
3 Express in scientific notation: a 0.035 b 0.000 076 e 0.000 003 f 0.913 i 0.000 001 j 0.89
c 0.8 g 0.000 007 146 k 0.000 000 078
4 Express in scientific notation: a 25 000 d 7 g 0.000 875 j 0.000 000 000 9 m 0.552 p 563.7 s 270 000 000
b e h k n q t
55 000 000 59 670 763 9057.6
4 400 000 0.4 6.7 73 2299 0.0001 400.4
d h l p
c f i l o r u
5 Express each of the following in decimal form. b 5.7 × 10 2 c 5.7 × 101 a 5.7 × 10 3 − 1 − 2 e 5.7 × 10 f 5.7 × 10 g 5.7 × 10 −3 1 0 i 8 × 10 j 8 × 10 k 8 × 10 −1
95 15 24.7 130.2
d 0.0713 h 0.009 l 0.1 185 000 000 0.027 20 345 000 000 0.06 3 500 000 7.03 50
Example 25
d 5.7 × 10 0 h 8 × 10 2 l 8 × 10 −2
6 Express each of the following in decimal form. a 6 × 105 b 7.1 × 10 3 d 3.14 × 100 e 6 × 10 −5 − 8 g 3.02 × 10 h 5.9 × 10 −10 − 4 j 4 × 10 k 5 × 10 3 − 1 m 8.03 × 10 n 6.32 × 10 4 − 7 p 2.2 × 10 q 9.0 × 10 6
c f i l o r
3.02 × 10 8 7.1 × 10 −3 1.1 × 10 12 4.76 × 10 − 4 1.6 × 10 −2 1.11 × 10 −1
7 Express in scientific notation: a two d four thousand g seven hundredths j fifteen hundred
c f i l
seven hundred three tenths fifteen hundredths six thousandths
8 Find the missing power: a 57.3 = 5.73 × 10? d 17 000 000 000 = 1.7 × 10? g 3.152 × 10? = 3.152
b e h k
ninety five million five millionths three hundred thousand b 8 = 8 × 10? e 4.3 × 10? = 430 h 1.128 × 10? = 0.000 1128
SkillBuilder 11-18 Scientific notation for other numbers
c 0.000 004 = 4 × 10? f 7.5 × 10? = 0.75 i 9.05 × 10? = 905 000
Worksheet 7-07 Scientific notation puzzle
Just for the record ‘Big’ numbers The numbers 1000 and 1 000 000 have the names thousand and million, but what about the names of numbers such as 1 000 000 000 and 1 000 000 000 000? The table below lists the names of some big numbers and their meanings.
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Name
Numeral
million
106 = 1 000 000
billion
109 = 1 000 000 000
trillion
1012
quadrillion
1015
quintillion
1018
sextillion
1021
septillion
1024
octillion
1027
nonillion
1030
decillion
1033
According to the Guinness Book of Records, the largest number for which there is an accepted name is the centillion, first recorded in 1852. It is equal to 10303. Find the name of the number that is equal to 10100.
Working mathematically Apply strategies and reasoning: On the blink (Work in pairs.) In common with all crabs, the blue swimmer crab (pictured) cannot blink.
1 Copy the following table. Trial 1
Trial 2
Average blinks per minute
name name 2 Get comfortable and relax. Have a partner watch your eyes and count how often you blink in a minute. Do this twice. Record your results in the table and find the average. 3 Repeat the experiment with you observing your partner. Record your results.
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4 Calculate how often you each blink in the following period. (Write your answers in scientific notation, correct to two significant figures.) a an hour b a waking day of 16 hours c a year d an average lifetime of 75 years 5 If a blink takes approximately 0.5 seconds, calculate how long you will each have your eyes closed in the following periods. a a minute b an hour c a waking day d a waking year e an average waking lifetime
Comparing numbers in scientific notation Example 26 Two soft-drink manufacturers each claim that their brand is the most popular in the world, based on last year’s sales. MAXI KOKE sold 5.2 × 1010 cans and KOLA FREE sold 7.9 × 109 cans. Who do you think sold the most?
Solution We can check our choice by writing each as an ordinary numeral. MAXI KOKE: 5.2 × 1010 = 52 000 000 000 KOLA FREE: 7.9 × 109 = 7 900 000 000 Since 52 000 000 000 > 7 900 000 000, we can say that MAXI KOKE had the most sales. Note that because the powers of ten are different, they are more important than the 7.9 or the 5.2 in making comparisons.
To compare numbers in scientific notation, first compare the powers of ten. If the powers of ten are the same, then compare the numbers between 1 and 10 that are multiplying the powers of ten.
Exercise 7-11 1 Choose the largest number from each of the following pairs. a 6 × 108 or 8 × 108 b 4.8 × 103 c 8.4 × 10 0 or 1.3 × 107 d 3.6 × 10 −7 9 9 e 9.3 × 10 or 7.6 × 10 f 3.5 × 10 − 6 0 − 4 g 3.04 × 10 or 3.04 × 10 h 4.5 × 10 −5 i 2 × 10 −15 or 2 × 10 −17 j 6.23 × 10 −2
Example 26
or or or or or
2.7 × 6.3 × 10 −7 9.3 × 102 3.7 × 10 −7 9.7 × 10 −2 10 5
Spreadsheet 7-04 Comparing scientific notation
2 Write in order: a 6 × 10 5, 6 × 10 2, 6 × 10 3 (smallest to largest) b 3.8 × 10 9, 7.3 × 10 9, 5.5 × 10 9 (largest to smallest) c 3 × 10 − 4, 3 × 10 − 6, 3 × 10 −5 (lowest to highest) d 4.1 × 10 −3, 9.5 × 10 −3, 6.4 × 10 −3 (highest to lowest)
Spreadsheet 7-05 Ordering scientific notation
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e 3.5 × 100, 5.3 × 10 4, 4.9 × 10 − 4 (ascending order) f 2.1 × 10 −8, 6.9 × 10 −1, 4.3 × 10 − 4 (descending order) g 5 × 10 −9, 6.3 × 102, 8 × 10 − 4, 9.76 × 10 (descending order) 3 The following table contains the approximate populations and areas of 10 countries. Population
Area (km2)
Australia
2.0 × 10 7
7.69 × 10 6
Cambodia
1.3 × 10 7
1.81 × 10 5
China
1.3 × 10 9
9.57 × 10 6
Indonesia
2.2 × 10 8
1.90 × 10 6
Japan
1.3 × 10 8
3.78 × 10 5
Lebanon
4.0 × 10 6
1.02 × 10 4
New Zealand
3.8 × 10 6
2.72 × 10 5
South Africa
4.5 × 10 7
1.22 × 10 6
Tonga
9.9 × 10 4
6.49 × 10 2
Vietnam
8.0 × 10 7
3.32 × 10 5
Country
a List the countries in descending order of population size. b List the countries in ascending order of area.
Just for the record Hair facts There are about 1.1 × 105 hairs on your head. Each hair grows at the rate of about 1.3 × 10 −3 cm per hour. A single hair lasts about six years. Every day you lose between 30 and 60 hairs. Each hair grows from a small depression in the skin called a follicle (a gland). After the hair falls out, the follicle rests for about three to four months before the next hair starts growing. Hair follicles are either oval, flat or round in shape. How straight, wavy or curly your hair is depends on the shape of your hair follicles. How many hairs are on all the heads in China?
Curly hair ■ flat follicle
256
Wavy hair ● oval follicle
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Straight hair ● round follicle
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Working mathematically Communicating and reasoning: Calculator displays Scientific notation can be entered and displayed on the calculator. A calculator display of 3.507 means 3.5 × 107. 1 Enter the following numbers on your calculator (using the EXP key) and then write down the calculator displays. a 4.7 × 10 9 b 3.56 × 10 15 c 6.7 × 10 − 6 − 10 − 4 d 4.2 × 10 e 2.047 × 10 f 9.8 × 10 23 Compare your results with those of other students. 2 Write down these calculator displays in scientific notation: a 2.7 11 b 4.02-05 c 8.7509
d
1.19-12
3 Dale and Amy were asked to evaluate 716 to two significant figures. Dale wrote down the answer as 3.313, while Amy wrote the answer as 3.3 × 1013. Which answer is correct? Explain. 4 Explain the difference between the numerical expressions 5 × 107 and 57. Compare your work with that of other students in your class. 5 When entering large or small numbers in a spreadsheet on a computer, E-notation is used. The computer display 6.2E+11 means 6.2 × 1011. Write down these computer displays in scientific notation: a 3.5E+18 b 6.2E–7 c 4.29E–13
Calculations in scientific notation Calculations using the index laws Scientific notation uses powers of ten, so the index laws can be used to evaluate questions that involve numbers in scientific (or standard) notation.
Example 27 Use index laws to simplify the following. Give your answers in scientific notation. a (3 × 10 4) × (8 × 107) b (1.2 × 10 7) ÷ (3 × 10 3) c (5 × 10 4)2
d
4 × 10
10
Solution
a (3 × 104) × (8 × 107) = (3 × 8) × (104 × 107) = 24 × 1011 = 2.4 × 1012 0.4
7
1.2 × 10 b (1.2 × 107) ÷ (3 × 10 3) = ---------------------3 1 3 × 10 = 0.4 × 10 4 = 4 × 10 3
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c (5 ×
10 4)2
= × = 25 × 108 = 2.5 × 109 52
(10 4)2
4 × 10
d
10
1 --2
1 --10 2
= 4 × ( 10 ) = 2 × 10 5
Using the calculator with scientific notation The calculator may also be used to evaluate expressions involving scientific notation. To enter scientific notation into the calculator, you need to use the EXP key.
Example 28 1 Enter each of the following into your calculator. a 6.2 × 1012 b 1.35 × 10 −3
Solution
a 6.2 × 1012 Enter: 6.2 EXP 12 The calculator display will be 6.212 , which means 6.2 × 1012 (or 6.2 × 1012 ). b 1.35 × 10 −3 Enter: 1.35 EXP
(–) 3
=
The calculator display will be 1.35-03 (or 1.35 × 10-03 ). 2 Calculate: a (4.25 × 107) × (8.2 × 106)
b (1.08 × 10 −15) ÷ (3 × 1011)
c (4.9 × 107)2
Solution a Enter: 4.25 EXP 7
×
=
8.2 EXP 6
Calculator display is 3.48514 (or 3.485 × 1014 ) . ∴ (4.25 × 107) × (8.2 × 106) = 3.485 × 1014 b Enter: 1.08 EXP
(–)
15
÷
3 EXP 11
=
Calculator display is (or 3.6 × ). ∴ (1.08 × 10 −15) ÷ (3 × 1011) = 3.6 × 10 −27 3.6-27
c Enter: 4.9 EXP 7
x2
10-27
=
Calculator display is 2.40115 (or 2.401 × 1015 ). ∴ (4.9 × 107)2 = 2.401 × 1015
Example 29 Evaluate each of these, giving your answers in scientific notation, correct to three significant figures. a (5.7 × 10 5) × (3.42 × 107) b (8.2 × 107)3
Solution
a (5.7 × 10 5) × (3.42 × 107) = 19.494 × 1012 ≈ 1.95 × 1013
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b (8.2 × 107)3 = 5.513 68 × 10 23 ≈ 5.51 × 10 23
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Example 30 Estimate each of the following. a (4.7 × 10 5) × (3.2 × 108)
b (8.4 × 1012) ÷ (1.93 × 107)
Solution (4.7 × 10 5) × (3.2 × 108) ≈ (5 × 10 5) × (3 × 108) = 15 × 1013 = 1.5 × 1014
a
(8.4 × 1012) ÷ (1.93 × 107)
b
8 × 10 12 ≈ -------------------2 × 10 7 = 4 × 10 5
Exercise 7-12 1 Use the index laws to simplify the following. Give your answers in scientific notation. a (2 × 103) × (3 × 10 5) b (8 × 107) ÷ (4 × 10 2) c e g i k
(2 × 10 5)3 (4 × 107) × (6 × 108) (4 × 103)5 (2 × 10 −3)2 (5 × 108) × (2 × 10 2)3
d f h j l
Example 27
Worksheet 7-08 Scientific notation problems
12
9 × 10 (1 × 108) ÷ (2 × 103) (9 × 10 5) × (8 × 103) ÷ (4 × 10 2) (9 × 10 − 4) ÷ (3 × 108) (4.2 × 10 5) ÷ (6 × 10 −5)
2 Find the answers to the following in scientific notation. a (8.4 × 107) × (3.4 × 108) b (9.4 × 10 12) + (8.3 × 10 15) − 9 − 10 c (4.9 × 10 ) − (3.7 × 10 ) d (15.75 × 10 −3) ÷ (5 × 107) e 24.08 ÷ (8 × 10 8)
f
g (3.2 × 10 9)2
h
3.969 × 10 3
8 × 10
Example 28
19
-9
7.62 × 10 9 ------------------------j (7.6 × 10 3) × (4.5 × 10 5) ÷ (3 × 10 −8) 2 × 10 -4 3 Evaluate, giving answers in scientific notation, correct to three significant figures: b (2.03 × 1035) + (1.23 × 1034) a (5.12 × 10 5) × (8.3 × 107) 30 29 c (7.4 × 10 ) − (3.59 × 10 ) d (1.076 × 1017 ) ÷ (2.3 × 1011) i
e
6.6 × 10
27
g (8.17 × 1016)3 i
(7.05 × 103) ÷ (3.9 × 107)
f
(7.5 × 10 23) ÷ (3.3 × 10 −13)
h
3
j
2.69 × 10
Example 29
26
( 5.6 × 10 4 ) × ( 3.9 × 10 5 ) -----------------------------------------------------------( 2.3 × 10 7 )
4 Estimate each of the following, leaving your answers in scientific notation. a (5.7 × 103) × (2.3 × 10 5) b (8.4 × 10 5) × (3.7 × 107) 20 5 d (1.6 × 10 8)2 c (9.1 × 10 ) ÷ (3.2 × 10 ) e (7.13 × 1010) × (9.8 × 10 8) f (1.99 × 10 11) ÷ (2.01 × 107) 4 8 g (5.85 × 10 ) ÷ (2.05 × 10 ) h (6.3 × 1012) ÷ (2.9 × 10 3)
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5 Manal’s answer to (8.3 × 1015) × (5.125 × 1017) was 4.25 × 1034, correct to three significant figures. a Estimate an answer to the calculation. b Is Manal’s answer correct? Give reasons. 6 Simplify, giving your answers in scientific notation correct to two significant figures: a 595 × 959 b 1000 ÷ 3 c 220 d 6 ÷ 11 e 81−1 f 3−10 g 99 h (0.75)−5 7 a The human body consists of approximately 6 × 10 9 cells, and each cell consists of 6.3 × 10 9 atoms. Roughly how many atoms are there in a human body? (Express your answer in scientific notation.) b The Earth is 1.52 × 10 8 km from the Sun and the speed of light is 3 × 10 5 km/s. How long does it take for light to travel from the Sun to Earth? Express your answer in: i seconds ii minutes. 8 Answer the following in scientific notation, correct to two significant figures where necessary. a A telephone directory is 4.5 cm thick. There are 2000 pages in it. Find the thickness, in millimetres, of one page. b The Sun burns 6 million tonnes of hydrogen a second. Calculate how many tonnes of hydrogen it burns in a year (that is, 365 1--- days). 4 c Sound travels at approximately 330 metres per second. If Mach 1 is the speed of sound, how fast is Mach 5? Convert your answer to kilometres per second. d The distance light travels in one year is called a light year. If the speed of light is approximately 3 × 10 5 km per second, how far does light travel in one year? e The nearest star (Alpha Centauri) is 4.3 light years away from Earth. How long would it take a spaceship travelling from Earth at the speed of light to reach the star? f In a science fiction space movie, Warp 1 is the speed of light. If a starship travels at Warp 9, which is 9 times the speed of light, how fast is this in metres per second? g A thunderstorm is occurring 33 km from where you are standing. Use the speed of light (3 × 10 5 km per second) and the speed of sound (330 metres per second) to calculate: i how long the light from the lightning takes to reach you ii how long the sound from the thunder takes to reach you. 9 a What is the largest number that can be displayed on your calculator? b What is the smallest?
Working mathematically Applying strategies and reasoning: The reward for inventing chess (Or ‘How many grains of wheat on the chessboard?’) 1 On the first square of a chessboard there is 1 grain of wheat. On the second square there are 2 grains of wheat. On the third square there are 4 grains, on the fourth there are 8 grains, and so on. 2 Copy and complete this table. Number of square on the board Number of grains of wheat
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1st
2nd
3rd
4th
…
1
2
4
8
…
10th
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3 Look at the number of each square and the number of grains of wheat on it. What is the connection? Explain it in your own words. 4 How many grains of wheat would be on the: a 16th square? b 32nd square?
c 64th square?
5 If a single grain of wheat weighs 100 mg, how many tonnes of wheat would there be on the chessboard? 6 If the inventor had asked for 10c coins instead of grains of wheat, how much money would the king have had to pay?
Power plus 1 If p = 8, q = 4 and r = 25, evaluate each of the following. a p3 b q4 c r2 e q −2 i
(rq)3
2 Simplify: a 4y × 4y e (3a)2 i
1 --2
f
r
j
--q- p
g p -1
b 3e ÷ 3e f (32)a
m×m+m×m
d p −1
1 --3
h rq3
q -2 k --- p
l
c 10 x − 1 × 10 x + 1 g 5n × (5n)2
d 6x + 2 ÷ 6x h (8x)2 ÷ 8x 1 ---
p×p+p×p+p×p
j
pq2r
1 ---
1 ---
1 ---
k n2 × n2 + n2 × n2
3 Write the meaning of each of these: 1 ---
1 ---
a 16 4 2 ---
f
1 ---
b 81 4 2 ---
m5
1 ---
c 32 5 7 ------
g k3
3 ---
d 128 7
e x4
1 ---
h d 10
i
1 ---
32 5
yn
j
4 Simplify: 1 ---
a ( a8 ) 2 e
b 1 --20 2
( 100g )
f
16 p 8 1 --24 3
( 64h )
1 ---
c
3
8y 9
d
( 27y 15 ) 3
g
3
n 12
h
3
5 Write each of the following in scientific notation. a 94.2 × 10 9 b 0.52 × 10 −3 c 0.004 × 107 6 Write each of the following in scientific notation. a 6.7 million b 15.7 million d 4 billion e 3.2 billion
64h 15
d 105 × 10 −4
c 57.8 thousand f 127 million
7 Find one set of values for each of a, m and n that would make each of the following equations true. m ----
m ----
m ----
m ----
a an = 2 b an = 3 c a n = 64 d a n = 125 Present the results of the following activities in a written report of 1 to 2 pages.
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8 For how many values of a and b does a b = b a? 9 Investigate the solutions of the equation x n + y n = a n for various values of n. 10 The numbers 3, 5, 17, 257 and 65 537 can all be generated by a simple method, using the numbers 1 and 2 only. a What is this method? b What is the next number in the sequence? 11 On average: • the heart beats 70 times a minute • the skin sweats 0.3 L of liquid a day • the mouth eats 400 grams of food per meal • the lungs breathe 0.6 m3 of air an hour Calculate the total of each of these activities over a lifetime of 70 years (assuming three meals a day and 365 1--- days in a year). Express your answwers in scientific notation. 4
12 Investigate the sonic boom that occurs when an aircraft breaks the sound barrier. (Look up the key terms mach 1 and speed of sound.)
Language of maths base exponent integer reciprocal square root
Worksheet 7-09 Indices crossword
cube root fractional index negative index root zero index
EXP index power scientific notation
expanded form indices radical sign standard notation
1 Choose five words from the list. Use each word in a sentence to show that you understand its meaning. 2 Match each term in Column A with the correct term in Column B. A
B
negative power
EXP key
standard notation
reciprocal
exponent
square root, cube root
fractional index
index
3 The following mathematical words have different meanings when used in subjects other than mathematics. For each word, write two sentences, one using the word in its mathematical and one in its its non-mathematical sense. a base b index c power 4 Explain the difference between a base and a power. 5 What is scientific notation used for? 6 Fill in the missing letters: a _o_e_ d _n_e_
262
b _x__n_n_ e _ec_p___a_
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c _c_e_t_f__ f b_s_
07_NC_Maths_9_Stages_5.2/5.3 Page 263 Friday, February 6, 2004 2:17 PM
7 How many times can you find the word exponent hidden in the puzzle below? E XX PPP OOOO NNNNN E E E E E E NNNNNNN T T T T T T T T
Topic overview • Write 10 questions (with solutions) that could be used in a test for this chapter. Include some questions that you have found difficult to answer. • Swap your questions with another student and check their solutions against yours. • List the sections of work in this chapter that you did not understand. Follow up this work with a friend or your teacher. • Copy and complete the summary of this topic shown below. Have your overview checked by your teacher to make sure nothing is missing or incorrect.
1 2 3 4
Scientific notation
Zero index
• 4 × 102 = 400 • 3 × 10 −4 = 0.0003 • 9372 = 9.372 × 103
• 80 = 1 • 2340 = 1 • (1.1)0 = 1
Index laws am × an = am + n am ÷ an = am − n (a m)n = a mn (ab)n = a nb n
INDICES
a n an 5 - = ----n b b 6 a0 = 1
index or exponent or power
base
-a b
-1
b =-, a
-a b
-n
b n = - a
Negative powers • 7−1 = 1-7 3 −2 4 2 16 • -- = -- = ----4 3 9
Fractional powers 1 ---
1 ---
• 82 =
2 • 53 = 3 5 1 1 square root = power of --cube root = power of --2 3
1 - ---
1 --------- = 10 2 10
1 ---
an = n a m ----
m m a n = (n a) = n a
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Chapter 7 Ex 7-01
Ex 7-02
Ex 7-03
c 10 f m×n×n×n i 3×a×3×a×a
2 Simplify: a y3 × y10 d 3p2 × 2p5 g 5x5 × 3x3y
b a × a4 e 3q3 × 3q 8 h 10x2y × 3xy 2
c h8 × h 2 f 5m7 × 2m i 4ab4 × 5a3b
3 Simplify: a 420 ÷ 44 d 10e15 ÷ 5e3
b x8 ÷ x2 e 20n9 ÷ 4n3
c b12 ÷ b f 24g8 ÷ 3g2
36a 8 b 3 h ----------------4a 4 b
i
Ex 7-06
Ex 7-07
p6 q3 ----------p2 q2
4 Simplify: a (a2)4 e (5r 2)3 i
Ex 7-05
b (y5)5 f (4w 4)4
(10g)3
m 4 ------5 w
j
5 Simplify: a 990 d (−d)0 g 50 + (5m)0
5
1 ---
f
1 ---
b 64 3
c
1 --4 2
1 --( 2q ) 3
2a 7 4 k -------- e
l
or
3
1 --1 3
d m −5 i 3x −1 .
i
8 Simplify each of the following. a ( 125d )
1 --20 4
b ( 16y )
3 ---
9 Evaluate 40 5 , correct to two decimal places.
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e y −1
d ( q 15 ) 3
h ( xy ) 3
c
(3m3np 2)5
c d0 f p0q i 4 × (9p)0
1 ---
- -- 8
1 ---
g ( 4a )
4 --15 3
Ex 7-08
d (2x3)5 h (5a2b)2
b (−99)0 e (pq)0 h 15x0 + (15x)0
7 Write each of the following using either
100x 2 y 4 -------------------5xy 2
c (b)3 g (a2b)2
6 Write each of the following with a positive index. a 8 −1 b 8 −2 c (9m)−1 − 2 − 3 f y g y h (3x)−1
a 64 2
Ex 7-08
Topic test Chapter 7
1 Write each of the following in index notation. a 4×4×4 b 6×6 d x×x×x×x e y×y g 7×4×7×7×4 h a×b×b×a
g Ex 7-04
Review
2 --8 5
( 32x )
1 ---
e ( p 10 ) 2
1 ---
( 8k ) 2
2 ---
p 3 3 d ----- v6
07_NC_Maths_9_Stages_5.2/5.3 Page 265 Friday, February 6, 2004 2:17 PM
4 ---
10 The value of 27 3 is: A 243
11 The value of 32
3 - --5
Ex 7-08
B 81
C 9
D 3
is:
A −19.2
Ex 7-09
B 8
1 8
1 D --8
− ---
C
12 Simplify:
Ex 7-09
3 ---
a ( 16 p 4 ) 4 d ( 32h 10 )
2 - --5
4 --6 9 3
j
3
e
( -8x 9 )
3 ---
-64n 12 5 --3
2 --4 6 5
g ( 1000c d ) 4
b
h ( 32a b )
5 --12 3 3
8 -3
( 625a )
k ( -27d h )
c
( 625w 16 ) 4
f
( -27m 9 n 3 )
i
4
256k 8 m 4
l
5
32b 15
2 - --3
13 Evaluate each of the following. a 64
2 - --3
+4
1 - --2
Ex 7-09
b
( -32 )
3 --5
− ( -8 )
4 --3
3 --2
9 ÷ 36
c
3 - --2
14 Write each of the following in scientific notation. a 55 000 b 0.55 d 0.000 25 e 8
c 250 000 f 0.000 000 000 08
15 Write each of the following in decimal form. a 8.1 × 10 3 b 6 × 10 7 d 8.1 × 10 −3 e 6 × 10 −7
c 3.075 × f 3.075 × 10 −2
Ex 7-10
Ex 7-10
10 0
16 Arrange the numbers in each of these sets in ascending order: a 6.8 × 10 7, 3.5 × 10 7, 7.5 × 10 7 b 3 × 10 3, 9 × 10 − 8, 4 × 10 0 c 4.4 × 10 −3, 5.7 × 10 −7, 3.1 × 10 −1
Ex 7-11
17 Evaluate each of the following, giving your answers in scientific notation correct to two significant figures. a (3.65 × 10 −22) × (7.4 × 10 8) b (1.44 × 1010) ÷ (3.6 × 10 4)
Ex 7-12
1 ---
d (6.25 × 10 − 8) 2
c (5 × 10 5)3
18 Evaluate, giving your answers in scientific notation correct to two significant figures. a (3 × 10 −8)3 ÷ (2.8 × 10 −5) c (8.4 ×
10 3)−2
÷ (4.8 ×
107)
e 9068 ÷ (0.000 35)2 g
( 9.7 × 10 5 ) × ( 1.3 × 10 8 ) -----------------------------------------------------------2 ( 5.75 × 10 -3 )
( 7 × 10 8 ) × ( 3.4 × 10 5 )
b
d (5.64 × f
Ex 7-12
3
5 --20 10 ) 3
3.6 × 10
-9
× (8.1 × 103)2
1 h ----------------------------1.57 × 10 8 I NDI CE S
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