Download 3 • Index laws

number and algebra
TopIC 3
N
LY
Index laws
3.1 Overview
N
AL
U
AT
IO
Indices (the plural of index) are the lazy mathematician’s way of
abbreviating operations such as multiplication and division. You do not
have to use this technique if you have the time to do otherwise. For
example, the lazy mathematician might write x 6, whereas you could
write x × x × x × x × x × x .
Why use indices at all? Because doing so will save you time and
effort in more difficult calculations later. Will you use indices later in
life? That depends on what work you decide to do. However, if you
are contemplating any work associated with computers, cars,
engines, science, design or apprenticeships, then the answer is
almost certainly yes.
O
Why learn this?
What do you know?
M
PL
E
EV
1 THInK List what you know about index laws. Use a thinking
tool such as a concept map to show your list.
2 paIr Share what you know with a partner and then with
a small group.
3 SHare As a class, create a thinking tool such as a large concept
map that shows your class’s knowledge of index laws.
Learning sequence
SA
3.1 Overview
3.2 Review of index form
3.3 First Index Law (multiplying numbers in index form with the
same base)
3.4 Second Index Law (dividing numbers in index form with the
same base)
3.5 Third Index Law (the power of zero)
3.6 Fourth Index Law (raising a power to another power)
3.7 Review ONLINE ONLY
c03IndexLaws.indd 40
09/05/16 10:03 AM
N
LY
O
N
AT
IO
AL
U
EV
E
M
PL
SA
c03IndexLaws.indd 41
09/05/16 10:03 AM
number and algebra
3.2 Review of index form
WorKed eXample 1
AT
IO
State the base and power for the number 514.
N
O
N
LY
• If a number or a variable is multiplied by itself several times, it can be written using
short-cut notation referred to as index form.
• When written in index form, the number or the variable that is being multiplied is written
once only. To indicate how many times it is being multiplied by itself, a small number
is written above and to the right of it. For example, if number 2 is multiplied by itself
4 times, it can be written as 2 × 2 × 2 × 2 (factor form), or as 24 (index form).
• When written in index form, the number or variable that is being multiplied is called the
base, while the number showing how many times it is being multiplied is called the power,
or index. For example, in the number 24, 2 is the base and 4 is the power or index.
• When the base is multiplied by itself the number of times indicated by the power, the
answer is called a basic numeral. For example,
24
=2×2×2×2=
16
Index form
Factor form
Basic numeral
THInK
WrITe
514
Write the number.
2
The base is the number below the power.
The base is 5.
3
The power or index is the small number just
above and to the right of the base.
The power is 14.
EV
WorKed eXample 2
AL
U
1
Write 124 in factor form.
THInK
Write the number.
2
The base is 12, so this is what will be
multiplied.
3
The power is 4, so this is how many times 12
should be written and multiplied.
SA
M
PL
E
1
124
= 12 × 12 × 12 × 12
WorKed eXample 3
Write 2 × 5 × 2 × 2 × 5 × 2 × 5 in index form.
THInK
42
WrITe
WrITe
2×5×2×2×5×2×5
1
Write the problem.
2
Write the factors in numerical order.
=2×2×2×2×5×5×5
3
The number 2 has been written 4 times and
multiplied. The number 5 has been written
3 times and multiplied.
= 24 × 53
Maths Quest 8
c03IndexLaws.indd 42
09/05/16 10:03 AM
number and algebra
WorKed eXample 4
Write 7 × 53 × 65 in factor form.
THInK
WrITe
7 × 53 × 65
Write the problem.
2
List the factors: 7 is written once, 5 is
written 3 times and multiplied, and 6 is
written 5 times and multiplied.
=7×5×5×5×6×6×6×6×6
N
LY
1
IndIVIdual paTHWaYS
⬛
ConSolIdaTe
⬛ ⬛ ⬛ Individual pathway interactivity
FluenCY
4
7812
j d 42
e
doc-6834
EV
2
reFleCTIon
How will you remember the
meaning of the base and that
of the index?
int-4401
State the base and power for each of the following.
a 8
b 710
c 2011
d 190
f 3100
g m5
h c24
i n36
Write the following in index form.
a 2×2×2×2×2×2
b 4×4×4×4
c x×x×x×x×x
d 9×9×9
e 11 × l × l × l × l × l × l × l
f 44 × m × m × m × m × m
WE2 Write the following in factor form.
a 42
b 54
c 75
d 63
6
7
4
e 3
f n
g a
h k10
Write each of the following as a basic numeral.
a 35
b 44
c 28
d 113
e 74
f 63
g 110
h 54
a MC What does 63 mean?
a 6×3
b 6×6×6
d 6+6+6
e 3×6
5
b What does 3 mean?
a 3×5
b 5×5
d 3×3×3×3×3
e 5×3
WE1
maSTer
Questions:
1f–j, 2, 3e–h, 4e–h, 5, 6e–h, 7,
8a, c, f, 9–12, 14
AL
U
1
⬛
Questions:
1–9, 11–13
N
praCTISe
Questions:
1–8, 13
AT
IO
⬛
O
Exercise 3.2 Review of index form
4
5
SA
3
M
PL
E
doc-6835
C
3×3×3×3×3×3
C
3+3+3+3+3
Topic 3 • Index laws 43
c03IndexLaws.indd 43
09/05/16 10:03 AM
number and algebra
Write each of the following in index form.
6×2×2×4×4×4×4
7×7×7×7×3×3×3×3
19 × 19 × 19 × 19 × 19 × 2 × 2 × 2
13 × 13 × 4 × 4 × 4 × 4
66 × p × p × m × m × m × m × m × s × s
21 × n × n × 3 × i × i × i × 6 × r × r × r
16 × k × e × e × e × 12 × p × p
11 × j × j × j × j × j × 9 × p × p × l
a
b
c
d
e
f
g
h
Write each of the following in factor form.
a 15f 3j4 b 7k6s2
d 19a4mn3
e 8l4r2t2
7 WE4 4b3c5
O
c
N
LY
6 WE3 UNDERSTANDING
each of the following numbers as a product of its prime factors, using indices.
a 64 b 40 c 36
d 400 e 225 f 2000
9Some basic numerals (see below) are written as the product of their prime factors.
Identify each of these basic numerals.
a 23 × 3 × 5 b 22 × 52 c 23 × 33
d 22 × 7 × 11 e 32 × 52 × 7 f 26 × 54 × 19
10 a Write each of the following numbers in index form with base 10.
i 10 ii 100 iii 1000 iv 1 000 000
b Use your knowledge of place value to rewrite each of the following basic numerals
in expanded form using powers of 10. The first number has been done for you.
i
SA
M
PL
ii
E
EV
AL
U
AT
IO
N
8Write
c
Basic numeral
Expanded form
 230
2 × 102 + 3 × 101
 500
iii
 470
iv
2360
v
1980
vi
5430
Write each of the following as a basic numeral.
i 7 × 104 + 5 × 103
ii 3 × 104 + 6 × 102
iii 5 × 106 + 2 × 105 + 4 × 102 + 8 × 101
Reasoning
what a3b4 means. Write a3b4 as a basic numeral in factor form as part of your
explanation.
11 Explain
44 Maths Quest 8
c03IndexLaws.indd 44
09/05/16 10:03 AM
number and algebra
problem SolVIng
a, b and c are prime numbers.
a Write 8 × a × a × b × b × b × c × c × c × c as a product of prime factors in index
form.
b If a = 2, b = 3 and c = 7, calculate the value of the basic numeral represented by
your answer in part a.
13 a Rewrite the numbers 140 and 680 in expanded form using powers of 10.
b Add the numbers in expanded form.
c Convert your answer for part b into a basic numeral.
d Check your answer for part c by adding 140 and 680.
e Try the question again, this time with two numbers of your own. Choose numbers
between 1000 and 10 000.
14 a Rewrite the numbers 10 and 14 in expanded form with powers of 2 using, as
appropriate, 23, 22 and 21.
b Add the numbers in expanded form.
c Convert your answer for part b into a basic numeral.
d Check your answer for part c by adding 10 and 14.
e Try the question again, this time with two numbers of your own. Choose numbers
between 16 and 63.
AT
IO
N
O
N
LY
12
AL
U
3.3 First Index Law (multiplying numbers in index
form with the same base)
SA
M
PL
E
EV
• The numbers in index form with the same base can be multiplied together by being
written in factor form first. For example, 53 × 52 = (5 × 5 × 5) × (5 × 5) = 55.
• The simpler and faster way to multiply numbers or variables in index form with the
same base is to use the First Index Law. The First Index Law states: am × an = am + n.
This means that when numbers in index form with the same base are multiplied by
each other, the powers (indices) are added together. For example, 53 × 52 = 53 + 2 = 55
(as above).
• If the variables in index form that are being multiplied have coefficients, the coefficients are
multiplied together and the variables in index form are multiplied, and simplified using the
First Index Law. For example, 2a4 × 3a5 = (2 × 3) × (a4 × a5) = 6a9.
↑
Coefficients
multiplied
↑
Variables multiplied, which
means indices are added
WorKed eXample 5
Simplify 23 × 26 after first writing in factor form, leaving the answer in
index form.
THInK
WrITe
23 × 26
1
Write the problem.
2
Write in factor form.
= (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2 × 2)
3
Simplify by writing in index form.
= 29
Topic 3 • Index laws 45
c03IndexLaws.indd 45
09/05/16 10:03 AM
number and algebra
WorKed eXample 6
Simplify 74 × 7 × 73, giving your answer in index form.
THInK
WrITe
74 × 7 × 73
1
Write the problem.
2
Show that the 7 in the middle has an index of 1.
3
Check to see if the bases are the same. They are all 7.
4
Simplify by using the First Index Law (add indices).
= 74 × 71 × 73
N
LY
= 74 + 1 + 3
= 78
WorKed eXample 7
O
Simplify 5e10 × 2e3.
WrITe
N
THInK
5e10 × 2e3
Write the problem.
2
The order is not important when multiplying, so place
the numbers first.
= 5 × 2 × e10 × e3
3
Multiply the numbers.
= 10 × e10 × e3
4
Check to see if the bases are the same. They are both e.
5
Simplify by using the First Index Law (add indices).
AL
U
AT
IO
1
= 10e10 + 3
= 10e13
EV
Multiplying expressions containing numbers in index form
with different bases
M
PL
E
• When there is more than one variable involved in the multiplication question, the First
Index Law is applied to each variable separately.
WorKed eXample 8
Simplify 7m3 × 3n5 × 2m8 × n4.
SA
THInK
46
WrITe
7m3 × 3n5 × 2m8 × n4
1
Write the problem.
2
The order is not important when
multiplying, so place numbers first and
group the same variables together.
= 7 × 3 × 2 × m3 × m8 × n5 × n4
3
Simplify by multiplying the numbers and
using the First Index Law for bases that are
the same (add indices).
= 42 × m3 + 8 × n5 + 4
= 42m11n9
Maths Quest 8
c03IndexLaws.indd 46
09/05/16 10:03 AM
number and algebra
Exercise 3.3 First Index Law (multiplying
numbers in index form with the same base)
IndIVIdual paTHWaYS
praCTISe
⬛
Questions:
1, 2, 3a–f, 4, 5, 6a–f, 11
ConSolIdaTe
⬛
Questions:
1–7, 10, 11, 13
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
2g–l, 3g–l, 4, 5, 6f–j, 7–13
int-4402
FluenCY
1
reFleCTIon
The First Index Law can only
be applied if the bases are the
same. Why is that so?
Simplify the following after first writing in factor form.
× 24 = (2 × 2) × (□ × □ × □ × □)
= 2□
b 53 × 55 = (5 × 5 × 5) × (□ × □ × □ × □ × □)
= 5□
c f 6 × f × f 2 = (□ × □ × □ × □ × □ × □) × □ × (□ × □)
=f□
Simplify each of the following.
a 37 × 32
b 614 × 63
c 106 × 104
d 113 × 113
e 78 × 7
f 211 × 23
g 52 × 52
h 89 × 82
i 137 × 138
j q23 × q24
k x7 × x7
l e × e3
WE6 Simplify each of the following, giving your answer in index form.
a 34 × 36 × 32
b 210 × 23 × 25
c 54 × 54 × 59
d 68 × 6 × 62
e 10 × 10 × 104
f 172 × 174 × 176
g p7 × p8 × p7
h e11 × e10 × e2
i g15 × g × g12
j e20 × e12 × e6
k 3 × b2 × b10 × b
l 5 × d4 × d5 × d7
a MC What does 6 × e3 × b2 × b4 × e equal?
a 6b6e4
b 6b6e3
C 6b9e
d 6b10e
e 6b8e3
b What does 3 × f 2 × f 10 × 2 × e3 × e8 equal?
a 32e11f 12
b 6e11f 12
C 6e23f
d 6e24f 20
e 3e24f 12
WE7 Simplify each of the following.
a 4p7 × 5p4
b 2x2 × 3x6
c 8y6 × 7y4
d 3p × 7p7
e 12t3 × t2 × 7t
f 6q2 × q5 × 5q8
WE8 Simplify each of the following.
a 2a2 × 3a4 × e3 × e4
b 4p3 × 2h7 × h5 × p3
c 2m3 × 5m2 × 8m4
d 2gh × 3g2h5
e 5p4q2 × 6p2q7
f 8u3w × 3uw2 × 2u5w4
g 9dy8 × d3y5 × 3d7y4
h 7b3c2 × 2b6c4 × 3b5c3
i 4r2s2 × 3r6s12 × 2r8s4
j 10h10v2 × 2h8v6 × 3h20v12
WE5
doc-2160
5
6
SA
M
PL
4
E
EV
3
AL
U
2
AT
IO
N
O
a 22
N
LY
⬛
Topic 3 • Index laws 47
c03IndexLaws.indd 47
09/05/16 10:03 AM
number and algebra
underSTandIng
Simplify each of the following.
a 3x × 34
b 3y × 3y + 2
3
1
2
c 32y + 1 × 34y − 6
d 32 × 33 × 34
8 a Express the following basic numerals in index form: 9, 27 and 81.
b Use your answers to part a to help you simplify each of the following expressions.
(Give each answer in index form.)
i 34 × 81 × 9
ii 27 × 3n × 3n − 1
N
LY
7
reaSonIng
Explain why 2x × 3y does not equal 6(x+y).
10 Step 1: The prime number 5 is multiplied by itself n times.
Step 2: The prime number 5 is multiplied by itself m times.
Step 3: The answers from steps 1 and step 2 are multiplied together.
Explain how you arrive at your final answer. What is your answer?
AT
IO
N
O
9
problem SolVIng
One dollar is placed on a square of a chess board, two dollars on the next square,
four dollars on the next square, eight dollars on the next square and so on.
a Write the number of dollars on the 10th square in index form.
b Write the number of dollars on the rth square in index form.
c How much money is on the 6th and 7th squares in total?
d How much money is on the 14th and 15th squares in total? Write your answer in
index form.
e Simplify your answer to part d by first taking out a common factor.
f Compare the numerical value of the number of dollars on the 64th square of the
chess board to the numerical value of the distance in kilometres to the nearest star,
Proxima Centauri.
12 a If x2 = x × x, what does (x3)2 equal?
b If the sides of a cube are 24 cm long, what is the volume of the cube in index form?
(Hint: The volume of a cube of side length s cm is s3 cm3.)
c What is the side length of a cube of volume 56 mm3?
d What is the side length of a cube of volume (an)3p mm3?
13 If I square a certain number, then multiply the result by three times the cube of the
certain number before adding one, the result is 97. What is the certain number?
SA
M
PL
E
EV
AL
U
11
48
Maths Quest 8
c03IndexLaws.indd 48
09/05/16 10:03 AM
number and algebra
3.4 Second Index Law (dividing numbers in index
form with the same base)
• The numbers in index form with the same base can be divided by first being written in
factor form. For example:
26 2 × 2 × 2 × 2 × 2 × 2
26 ÷ 2 4 = 4 =
2×2×2×2
2
2×2×2×2×2×2
=2×2
2×2×2×2
= 22
• The simpler and faster way to divide the numbers in index form is to apply the Second
Index Law. The Second Index Law states: am ÷ an = am − n. This means that when the
numbers or variables in index form with the same base are divided, the powers are
subtracted. For example, 26 ÷ 24 = 26 − 4 = 22 (as above).
O
N
LY
=
510
53
1
Write the problem.
2
Write in factor form.
3
Cancel 5s.
4
Write in index form.
AT
IO
WrITe
5×5×5×5×5×5×5×5×5×5
5×5×5
AL
U
THInK
510
after first writing in factor form, leaving your answer in index form.
53
=
=5×5×5×5×5×5×5
WorKed eXample 10
EV
Simplify
N
WorKed eXample 9
= 57
M
PL
THInK
E
Simplify d12 ÷ d4 using an index law.
WrITe
Write the problem and express
it as a fraction.
d12 ÷ d4
2
Check to see if the bases are the
same. They are both d.
=
Simplify by using the Second
Index Law (subtract indices).
= d12 − 4
= d8
SA
1
3
d12
d4
Dividing with coefficients
• When the coefficients are present, we divide them as we would divide any other numbers
and then apply the Second Index Law to the variables.
• In examples where the cofficients do not divide evenly, we simplify the fraction that is
formed by them.
• When there is more than one variable involved in the division question, the Second Index
Law is applied to each variable separately.
Topic 3 • Index laws 49
c03IndexLaws.indd 49
09/05/16 10:03 AM
number and algebra
WorKed eXample 11
Simplify 36d7 ÷ 12d3 giving your answer in index form.
THInK
WrITe
1
Write the problem and express it as a fraction.
2
Divide the numbers (or coefficients) and apply
the Second Index Law to the variables.
36d7
12d3
3d7
= 3
d
3
Simplify by using the Second Index Law
(subtract indices).
= 3d7 − 3
= 3d4
N
LY
36d7 ÷ 12d3 =
7t3 × 4t8
.
12t4
N
Simplify
O
WorKed eXample 12
THInK
Write the problem.
2
Multiply the numbers in the numerator and apply the
First Index Law (add indices) in the numerator.
3
Simplify the fraction formed and apply the Second Index
Law for the variable (subtract indices).
AL
U
AT
IO
1
WrITe
7t3 × 4t8
12t4
28t11
=
12t4
=
7t7
3
EV
Exercise 3.4 Second Index Law (dividing
numbers in index form with the same base)
IndIVIdual paTHWaYS
praCTISe
E
⬛
Questions:
1–6, 7a, b, 14
M
PL
reFleCTIon
How will you remember
that when numbers in index
form are divided, powers are
subtracted but coefficients are
divided?
⬛
ConSolIdaTe
⬛
Questions:
1–8, 11, 12, 14
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1, 2g–l, 3g–l, 4, 5, 6e–i, 7–14
int-4403
SA
FluenCY
doc-6836
doc-6837
doc-6838
50
Simplify each of the following after first writing in factor form, leaving your
answer in index form.
108
77
25
a
b
c
22
73
105
2 WE10 Simplify each of the following using the index law, leaving your answer in
index form.
a 33 ÷ 32
b 119 ÷ 112
c 58 ÷ 54
d 126 ÷ 12
1013
e 345 ÷ 342
f 1375 ÷ 1374
g 623 ÷ 619
h
109
456
f 1000
15
b77
h78
i
j
k
l
h
b7
15423
f 100
1
WE9
Maths Quest 8
c03IndexLaws.indd 50
09/05/16 10:03 AM
number and algebra
3
Simplify each of the following, giving your answer in index form.
3x ÷ x3
b 6y7 ÷ y5
c 8w12 ÷ w5
34
30
12
3
12q ÷ 4q
e 16f ÷ 2f
f 100h100 ÷ 10h10
45p14
48g8
80j15 ÷ 20j5
h
i
9p4
6g5
81m6
100n95
12b7
k
l
8b
18m2
40n5
MC What does 21r20 ÷ 14r10 equal?
WE11
4 a
a
7r10
b
3r2
2
C
7r2
2m33
equal?
16m11
8
m22
a
b
C 8m22
8
m22
5 WE12 Simplify each of the following.
b
a
d
3r10
2
e
2 10
r
3
d
m3
8
e
None of the above
What does
15p12
b
5p8
18r6
3r2
60b7
100r10
e
20b
5r6
6 WE12 Simplify each of the following.
8p6 × 3p4
12b5 × 4b2
a
b
18b2
16p5
27x9y3
16h7k4
d
e
12xy2
12h6k
AL
U
EV
8p3 × 7r2 × 2s
6p × 14r
27a9 × 18b5 × 4c2
18a4 × 12b2 × 2c
25m12 × 4n7
15m2 × 8n
12j8 × 6f 5
f
8j3 × 3f 2
c
i
81f 15 × 25g12 × 16h34
27f 9 × 15g10 × 12h30
E
underSTandIng
h
45a5
5a2
9q2
f
q
c
d
g
N
LY
j
doc-2161
O
g
N
d
AT
IO
a
5
Simplify each of the following.
a 210 ÷ 2p
b 27e ÷ 23e − 4
54x × 53y
32 − 3m × 37m
c
d
52y × 5x
35m × 3
8 × 16 × 4
8 Consider the fraction
.
2 × 32
a Rewrite the fraction, expressing each basic numeral as power of 2.
b Simplify by giving your answer
i in index form
ii as a basic numeral.
c Now check your answer by cancelling and evaluating the fraction in the ordinary way.
6 × 27 × 36
9 Consider the fraction
.
12 × 81
a Rewrite the fraction, expressing each basic numeral as the product of its prime
factors.
b Simplify, giving the answer:
i in index form
ii as a basic numeral.
SA
M
PL
7
Topic 3 • Index laws 51
c03IndexLaws.indd 51
09/05/16 10:03 AM
number and algebra
reaSonIng
12x
does not equal 4(x−y).
3y
11 Step 1: The prime number 3 is multiplied by itself p times.
Step 2: The prime number 3 is multiplied by itself q times.
Step 3: The answer from step 1 is divided by the answer from step 2.
Explain how you arrive at your final answer. What is your answer?
10
Explain why
problem SolVIng
By considering ap ÷ ap, show that any base raised to the power of zero equals 1.
1
13 By considering m6 ÷ m8, show that m−2 =
.
m2
14 I cube a certain number, then multiply the result by six. I now divide the result by the
certain number to the power of five. The result is 216. What is the certain number?
AL
U
AT
IO
N
O
doc-6851
N
LY
12
3.5 Third Index Law (the power of zero)
Method 1
2×2×2
2×2×2
E
23 ÷ 23 =
EV
• Consider the following two different methods of simplifying 23 ÷ 23.
23
23
= 23−3 (using the Second Index Law)
= 20
23 ÷ 23 =
8
8
=1
As the two results should be the same, 20 must equal 1.
• Any base that has an index power of 0 is equal to 1.
• The Third Index Law states: a0 = 1. This means that any base that is raised to the power
of zero is equal to 1.
• If it is in brackets, any numeric or algebraic expression that is raised to the power of zero
is equal to 1. For example, (2 × 3)0 = 1, (2abc2)0 = 1.
M
PL
=
SA
Method 2
WorKed eXample 13
Find the value of 150.
THInK
52
1
Write the problem.
2
Any base with an index of zero is equal to one.
WrITe
150
=1
Maths Quest 8
c03IndexLaws.indd 52
09/05/16 10:03 AM
number and algebra
WorKed eXample 14
Find the value of (25 × 36)0.
THInK
WrITe
(25 × 36)0
Write the problem.
2
Everything within the brackets has an index of zero,
so the answer is 1.
=1
N
LY
1
WorKed eXample 15
THInK
WrITe
2
Only a has a power of zero, so replace it with a 1 and
simplify.
WorKed eXample 16
6m3 × 11m14
.
3m10 × 2m7
AL
U
Simplify
= 19e5 × 1
= 19e5
AT
IO
Write the problem.
N
19e5a0
1
O
Find the value of 19e5a0.
THInK
WrITe
6m3 × 11m14
3m10 × 2m7
Write the problem.
2
Multiply the numbers and apply the First Index Law in
both the numerator and denominator.
=
3
Divide the numbers and simplify using the Second
Index Law.
= 11m17 − 17
= 11m0
4
Simplify using the Third Index Law.
= 11 × 1
= 11
66m17
6m17
SA
M
PL
E
EV
1
Exercise 3.5 Third Index Law (the power of zero)
IndIVIdual paTHWaYS
⬛
praCTISe
Questions:
1–8, 12
⬛
ConSolIdaTe
⬛
Questions:
1–8, 10, 12
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1–7, 8f–j, 9–12
reFleCTIon
Explain how the Second and
Third Index Laws are
connected.
int-4404
Topic 3 • Index laws 53
c03IndexLaws.indd 53
09/05/16 10:04 AM
number and algebra
FLUENCY
Find the value for each of the following.
b 440 c f 0 2 WE14 Find the value of each of the following.
a (23 × 8)0 b 70 × 60 c (35z4)0 3 Find the value of each of the following.
a 4 × 30 b 90 + 11 c c0 − 10 4 WE15 Find the value of each of the following.
1
WE13 a 160
12m3k0 b
7c0 + 14m0 c
32g0 + 40h0 h0
d
(12w7)0
d
3p0 + 19
8k0
7j0
3p × 4d 0
N
LY
a
d
d
4d 0 × 9p2
6b2 × 5c0
16t0
f
g
h
2z0 × 6p
8y0
3s0
12q0
5 Find each of the following.
a e10 ÷ e10 b a12 ÷ a12 c (4b3)0 ÷ (4b3)0
d 84f 11 ÷ 12f 11 e 30z9 ÷ 10z9 f 99t13 ÷ 33t13
N
O
e
6
AT
IO
UNDERSTANDING
Simplify each of the following.
21p4
40f 33
a
b
21p4
10f 33
c
54p6q8
27p6q8
d
16p11q10
8p2q10
x4y2z11
3c5d3l9
24a9e10
7i7m6r4
f
g
h
x4yz11
21i7m3r4
16a9e6
12c2d3l9
MC You are told that there is an error in the statement 3p7q3r5s6 = 3p7s6. To make the
7a
statement correct, what should the left-hand side be?
A (3p7q3r5s6)0 B (3p7)0q3r5s6 C 3p7(q3r5s6)0 D 3p7(q3r5)0s6 e 3(p7q3r5)0s6
EV
8f 6g7h3 8f 2
= 2 . To make the
6f 4g2h
g
statement correct, what should the left-hand side be?
You are told that there is an error in the statement
8f 6 (g7h3) 0
M
PL
A
E
b
AL
U
e
SA
c
What does
A
8
(6) 0f 4g2 (h) 0
WE16 3
2
B
8(f 6g7h3) 0
(6f 4g2h) 0
C
6k7m2n8
equal?
4k7 (m6n) 0
3n8
B
2
C
8( f 6g7) 0h3
(6f 4) 0g2h
3m2
2
D
D
8f 6g7h3
(6f 4g2h) 0
3m2n8
2
E
one of the
N
above
e
one of the
N
above
Simplify each of the following.
a
2a3 × 6a2
12a5
b
e
9k12 × 4k10
18k4 × k18
f
i
8u9 × v2
2u5 × 4u4
j
3c6 × 6c3
9c9
2h4 × 5k2
20h2 × k2
9x6 × 2y12
5b7 × 10b5
25b12
p3 × q4
g
5p3
c
d
h
8f 3 × 3f 7
4f 5 × 3f 5
m 7 × n3
5m3 × m4
3y10 × 3y2
54 Maths Quest 8
c03IndexLaws.indd 54
09/05/16 10:04 AM
number and algebra
reaSonIng
Explain why x0 = 1.
20 x 2
10 Simplify
, explaining each step of your method.
22 x 0
9
problem SolVIng
I raise a certain number to the power of three, then multiply the answer by three to the
power of zero. I then multiply the result by the certain number to the power of four
and divide the answer by three times the certain number to the power of seven. If the
final answer is three multiplied by the certain number squared, find the certain number.
Show each line of your method.
64x6y6z3
12 A Mathematics class is asked to simplify
. Peter’s answer is 4x3. Explain
2
6
3
16x y z
why this is incorrect, pointing out Peter’s error. What is the correct answer? Can you
identify another source of possible error involving indices?
O
N
LY
11
AT
IO
N
3.6 Fourth Index Law (raising a power to
another power)
E
EV
AL
U
• The Fourth Index Law states that when raising a power to another power, the indices are
multiplied; that is, (am)n = am × n. For example, (53)2 = 53 × 2 = 56.
• Every number and variable inside the brackets should have its index multiplied by the
power outside the brackets. That is,
(a × b) m = am × bm
a m am
= m
b
b
(These are sometimes called the Fifth and Sixth Index Laws.)
• Any number or variable that does not appear to have an index really has an index of one;
that is, 2 = 21, a = a1.
• Every number or variable inside the brackets must be raised to the power outside the
brackets. For example, (3 × 2)4 = 34 × 24 and (2a4)3 = 23 × a4 × 3 = 8a12.
M
PL
WorKed eXample 17
Simplify the following, leaving answers in index form.
(74)8
b
32
53
SA
a
3
THInK
a
b
1
Write the problem.
2
Simplify using the Fourth Index Law
(multiply the indices).
1
Write the problem.
2
Multiply the indices.
3
Simplify.
WrITe
a
b
(74)8
= 74 × 8
= 732
32 3
53
32 × 3
53 × 3
36
= 9
5
=
Topic 3 • Index laws 55
c03IndexLaws.indd 55
09/05/16 10:04 AM
number and algebra
WorKed eXample 18
Simplify (2b5)2 × (5b8)3.
THInK
WrITe
(2b5)2 × (5b8)3
Write the problem.
2
Simplify using the Fourth Index Law.
= 21 × 2b5 × 2 × 51 × 3b8 × 3
= 22b10 × 53b24
3
Calculate the coefficient.
= 4b10 × 125b24
= 500b10 × b24
4
Simplify using the First Index Law.
= 500b34
O
N
LY
1
WorKed eXample 19
N
2a5 3
.
d2
AT
IO
Simplify
THInK
WrITe
2a5
d2
3
Write the problem.
2
Simplify using the Fourth Index Law for each
term inside the grouping symbols.
=
3
Calculate the coefficient.
=
EV
AL
U
1
21 × 3a 5 × 3
d2 × 3
23a15
= 6
d
8a15
d6
M
PL
E
Exercise 3.6 Fourth Index Law (raising a power
to another power)
IndIVIdual paTHWaYS
SA
reFleCTIon
How will you remember to
raise all coefficients to the
power outside the brackets?
⬛
praCTISe
Questions:
1–4, 5a–i, 6a–c, 11, 12
⬛
ConSolIdaTe
⬛
Questions:
1–7, 9, 11, 12
⬛ ⬛ ⬛ Individual pathway interactivity
maSTer
Questions:
1d–i, 2d–i, 3d–i, 4, 5g–i, 6, 7i–o,
8g–o, 9–11, 13
int-4405
FluenCY
1
Simplify each of the following, leaving your answers in index form.
a (3 )
b (68)10
c (1125)4
d (512)12
e (32 × 103)4
f (13 × 173)5
WE17
2 3
g
56
33
22
10
h
(3w9q2)4
i
7e5
r2q 4
2
Maths Quest 8
c03IndexLaws.indd 56
09/05/16 10:04 AM
number and algebra
Simplify each of the following.
a (p ) × (q3)2
b (r5)3 × (w3)3
d (j6)3 × (g4)3
e (q2)2 × (r4)5
g (f 4)4 × (a7)3
h (t5)2 × (u4)2
WE18
(b5)2 × (n3)6
f (h3)8 × (j2)8
i (i3)5 × (j2)6
4 2
3
c
Simplify each of the following.
a (23)4 × (24)2
b (t7)3 × (t3)4
d (b6)2 × (b4)3
e (e7)8 × (e5)2
g (3a2)4 × (2a6)2
h (2d7)3 × (3d2)3
(a4)0 × (a3)7
f (g7)3 × (g9)2
i (10r12)4 × (2r3)2
c
What does (p7)2 ÷ p2 equal?
a p7
b p12
C p16
4 a MC
b
p11
e
w6p19
equal?
(wp)6
C
w14p36
d
w2p2
C
r8
d
r10
r3
b r4
5 Simplify each of the following.
a (a3)4 ÷ (a2)3
b
c (n5)3 ÷ (n6)2
d
7
3
2
2
e (f ) ÷ (f )
f
9
3
6
3
g (p ) ÷ (p )
h
(c6) 5
(c5) 2
j
k
(k3) 10
(k2) 8
l
(p12) 3
r12
int-2360
(p10) 2
3b4
d3
c
2k5
3t8
e
5y7
3z13
2
3
3
SA
a
E
Simplify each of the following.
M
PL
WE19
(f 5) 3
(f 2) 4
doc-2162
EV
i
(m8)2 ÷ (m3)4
(b4)5 ÷ (b6)2
(g8)2 ÷ (g5)2
(y4)4 ÷ (y7)2
e
AT
IO
What does (r6)3 ÷ (r4)2 equal?
a
6
e
O
(w2) 2 × (p3) 5
w2p6
a
c
(w5) 2 × (p7) 3
p4.5
N
What does
d
AL
U
b
N
LY
2
b
d
f
2
5h10
2j2
7p9
8q22
4a3
7c5
2
4
underSTandIng
7
Simplify each of the following using the index laws.
a g3 × 2g5
b 2p6 × 4p2
6
d 12x ÷ 2x
e (2d 3)2
g
15s8 ÷ 5s2
h
j
( f 4g3)2
k
m
5a6b2 × a2 × 3ab3
4bc6 × 3b3 × 5c2
16u6v5
6u3v
n x2y4 ÷ xy3
(w3)6
f 5a6 × 3a2 × a2
14x8
i
7x4
c
l
x2y4 × xy3
o
(4p2q5)3
Topic 3 • Index laws 57
c03IndexLaws.indd 57
09/05/16 10:04 AM
NUMBER AND ALGEBRA
REASONING
Simplify each of the following, giving your answer in index form. Justify your answer
in each case.
4x5 × 3x
a (w3)4 ÷ w2
b
c (2a3)2 × 3a5
4
2x
(2k3) 2
d 12x6 × 2x ÷ 3x5
e 2d 3 + d 2 + 5d 3
f
4k4
5
4p
g
h 15s8t3 ÷ 5s2t2 × 2st4
i 12b4c6 ÷ 3b3 ÷ 4c2
4
p × 6p
(3p3) 2 × 4p7
j (f 4g3)2 − fg3 × f 7g3
k
l 2(x2y)4 × 8xy3
2(p4) 3
4p2q7 × (3p3q) 2
m 5a6b2 + a2 × 3a4b2
n 24x2y4 ÷ 12xy3 − xy
o
6(pq) 3 × p5q4
b
c
0
9 Explain why ((a ) ) = 1.
10 A Mathematics class is asked to simplify (r4)3 ÷ (r3)2. Karla’s answer is r. Explain why
this is incorrect, pointing out Karla’s error. What is the correct answer?
SA
M
PL
E
EV
AL
U
AT
IO
N
O
N
LY
8
PROBLEM SOLVING
(22)
Show that 2(2 ) is not equal to (((2)2)2)2.
12 Arrange these numbers in ascending order.
11
2
doc-6852
58
22
3
2
5
32 , 22 , 23 , 52, 25 , 22
13 a Identify as many different expressions as possible that when raised to a power will
result in 16x8y12.
b Identify as many different expressions as possible that when raised to a
power will result in 312na6nb12n.
Maths Quest 8
c03IndexLaws.indd 58
09/05/16 11:01 AM
number and<STrand>
algebra
0.00
Review
3.7 Review
www.jacplus.com.au
Language
expanded form
factor form
index
index form
index laws
variable
M
PL
int-3183
E
EV
int-2620
AL
U
int-2619
base
basic numeral
coefficient
Download the Review
questions document
from the links found in
your eBookPLUS.
AT
IO
Theusing
Review
the contains:
most appropriate methods
• Fluency
questions
— allowing
demonstrate
the
problem Solving questions
—students
allowing to
students
to
skills
they
have
developed
to
effi
ciently
answer
questions
demonstrate their ability to make smart choices, to model
using
the most appropriate
methods
and investigate
problems, and
to communicate solutions
• problem
effectively.Solving questions — allowing students to
demonstrate
their
make smart
A summary
of the
keyability
pointstocovered
and achoices,
conceptto model
and
investigate
problems,
and
to
communicate
map summary of this topic are available as digital solutions
effectively.
documents.
Review
questions
N
LY
The Review contains:
The
Maths Quest
Review
is available
in a customisable
format
• Fluency
questions
— allowing
students
to demonstrate
the
for skills
students
to
demonstrate
their
knowledge
of
this
topic.
they have developed to efficiently answer questions
O
The Maths Quest Review is available in a customisable format
www.jacplus.com.au
for students
to demonstrate their knowledge of this topic.
N
ONLINE
ONLINE ONLY
ONLY
SA
Link to assessON for
questions to test your
readiness For learning,
your progress aS you learn and your
levels oF achievement.
Link
to assessON
Voluptin
quia for
volutem
assessON
provides
sets ofcupta
questions
every
vit
harunti
se course,
pario doassessON
VoassessON
topic
in your
as well as giving
instant
Voluptin
quia volutem
vit harunti
pario
feedbackcupta
and worked
solutions
to helpse
improve
dolorporplorporro
dollitae nia sim rempell
your mathematicalquia
skills.
orroreiumquo
core
volum
www.assesson.com.aufugit voluptat.
www.assesson.com.au
Link to SpyClass, an exciting
exciting
online
game
online game
combining
a
combining
a comicstory with
comic book–style
book–style
storylearning in
problem-based
with
problem-based
an immersive
environment.
learning
in
an
immersive
environment.
Join Jesse, Toby and Dan
and help
Link
assessON
themtotoptin
tackle
some ofassessON
the world’s
Voluptin
cupta quia
volutembyvitusing
harunti
most dangerous
criminals
the
se
pario dolorpharunti
se pariore
knowledge
you’ve gained
throughvolum
fugit
your voluptat.
study of mathematics.
www.spyclass.com.au
Topic 3 • Index laws 59
c03IndexLaws.indd 59
09/05/16 10:04 AM
number
<InVeSTIgaTIon>
InVeSTIgaTIon
and algebra
For rICH TaSK or <number and algebra> For puZZle
rICH TaSK
SA
M
PL
E
EV
AL
U
AT
IO
N
O
N
LY
Scientific notation and
standard form
60
Maths Quest 8
c03IndexLaws.indd 60
09/05/16 10:05 AM
N
Note: SI is the abbreviation for International System of Units.
O
N
LY
number
number and
and algebra
algebra
AL
U
AT
IO
Your calculator will accept very large or very small numbers when they are entered because it uses scientific
notation. Use the following steps to write the number 825 460 in scientific notation.
Step 1: Place a decimal point so that the number appears to be between 1 and 10.
8.254 60
Step 2: Count how many decimal places the decimal point is from its old position. (For whole numbers, this
is at the right-hand end of the number.) In this case, it is five places away.
8.254 60
Step 3: Multiply the number in step 1 by the power of 10 equal to the number of places in step 2.
8.254 60 × 105
EV
Note: If your number was made smaller in step 1, multiply it by a positive power to increase it
to its true value. If your number was made larger in step 1, multiply it by a negative power to
reduce it to its true value.
SA
M
PL
E
Proxima Centauri, near the Southern Cross, is the closest star to Earth and is 4.2 light-years away. A lightyear is the distance that light travels in 1 year. Light travels at 300 000 kilometres per second.
1 Write 300 000 km/s in scientific notation.
2 Find the distance travelled by light in 1 minute.
3 Find the distance travelled by light in 1 hour.
4 Find the distance travelled by light in 1 day.
5 Multiply your answer in question 4 by 365.25 to find
the length of a light-year in kilometres. (Why do we
multiply by 365.25?) Write this distance in scientific
notation.
6 Calculate the distance from Earth to Proxima
Centauri in kilometres.
7 Calculate the distance from Earth to some other
stars in both light-years and kilometres.
Topic 3 • Index laws 61
c03IndexLaws.indd 61
09/05/16 10:05 AM
<InVeSTIgaTIon>
number
and algebra
For rICH TaSK or <number and algebra> For puZZle
Code puZZle
N
LY
The largest gland
in the body
C
A
(12c 20) ÷ (2c 13)
(16c 9) ÷ (8c 5) =
=
7c 8 ÷ 7c 7
S
SA
=
O
8k 6e 2
4e 2k 4
=
E
3ct 4ec
M
PL
9c 2t 7e
EV
D
K
28f 10
4f 8
=
I
(6n 7u 7) ÷ (2n 5u 6)
a 60
10e 3k 12
18n7u12
3n 3u
=
a 20
=
T
B
2nu 8
=
2k5e3
=
R
35f 21 ÷ 7f 12
6e 10 ÷ 3e 2 =
8n 6u 11
N
=
W
H
L
e 10 ÷ e 7 =
P
=
AL
U
F
=
=
V
a6 ÷ a2 =
50t 12 ÷ 25t 11
62
8e 9c 15
18a 21 ÷ 3a 16
=
N
5f 4
24e 11c 15
=
AT
IO
15f 7
O
Simplify the expressions to find the puzzle’s answer code.
=
U
(20k 8t 6) ÷ (5k 6t 2k 2)
E
=
Y
Maths Quest 8
c03IndexLaws.indd 62
09/05/16 10:05 AM
NUMBER AND ALGEBRA
Activities
N
LY
O
3.7 Review
Interactivities
• Word search (int-2619)
• Crossword (int-2620)
• Sudoku (int-3183)
Digital docs
• Topic summary (doc-10755)
• Concept map (doc-10768)
• Topic review (Word doc-14928, PDF doc-14929)
AL
U
3.4 Second Index Law
Digital docs
• SkillSHEET (doc-6836) Equivalent fractions
• SkillSHEET (doc-6837) Simplifying algebraic
expressions
• SkillSHEET (doc-6838) Simplifying algebraic fractions
• Spreadsheet (doc-2161) Dividing with indices
• WorkSHEET 3.1 (doc-6851)
Interactivity
• IP interactivity 3.4 (int-4403) Second Index Law
3.6 Fourth Index Law
Digital docs
• WorkSHEET 3.2 (doc-6852)
• Spreadsheet (doc-2162) Raising a power to
another power
Interactivities
• Indices (int-2360)
• IP interactivity 3.6 (int-4405) Fourth Index Law
N
3.3 First Index Law
Digital doc
• Spreadsheet (doc-2160) Multiplying with indices
Interactivity
• IP interactivity 3.3 (int-4402) First Index Law
3.5 Third Index Law
Interactivity
• IP interactivity 3.5 (int-4404) Third Index Law
AT
IO
3.2 Review of index form
Digital docs
• SkillSHEET (doc-6834) Factor trees
• SkillSHEET (doc-6835) Squaring numbers
Interactivity
• IP interactivity 3.2 (int-4401) Review of index form
www.jacplus.com.au
SA
M
PL
E
EV
To access eBookPLUS activities, log on to
Topic 3 • Index laws 63
c03IndexLaws.indd 63
09/05/16 2:23 PM
number and algebra
ANSWERS
topic 3 Index laws
3.3 First Index Law (multiplying numbers in index form with
the same base)
3.2 Review of index form
AT
IO
N
O
N
LY
1 a 2 × 2 × 2 × 2 × 2 × 2 = 26
b 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 58
c f × f × f × f × f × f × f × f × f = f 9
2 a 39
b 617
c 1010
d 116
e 79
f 214
g 54
h 811
i 1315
j q47
k x14
l e4
3 a 312
b 218
c 517
d 611
e 106
f 1712
g p22
h e23
28
38
13
i g j e k 3b l 5d16
4 a A
b B
5 a 20p11
b 6x8
10
d 21p8
c 56y e 84t6
f 30q15
6 a 6a6e7
b 8h12p6 c 80m9
d 6g3h6
e 30p6q9 f 48u9w7
11 17
g 27d y h 42b14c9
i 24r16s18
38
20
j 60h v
23
7 a 3x + 4
b 32y + 2
c 36y − 5
d 312
2
3
4
8 a 9 = 3 ; 27 = 3 ; 81 = 3
b i 310
ii 32n + 2
9 The bases, 2 and 3, are different. The index laws do not apply.
10 Step 1: 5n
SA
Step 2: 5m
Step 3: 5n+m
11 a29
b2r − 1 c25 + 26 = $96
e213(1 + 2) = 3(213)
d213 + 214
f Check with your teacher.
12 a x6
b212
c52
d (an) p = anp
132
AL
U
M
PL
E
EV
1 a Base = 8, power = 4
b Base = 7, power = 10
c Base = 20, power = 11
d Base = 19, power = 0
e Base = 78, power = 12
f Base = 3, power = 100
g Base = m, power = 5
h Base = c, power = 24
i Base = n, power = 36
j Base = d, power = 42
2 a 26
b 44
c x5
3
7
d 9 e 11l f 44m5
3 a 4 × 4
b 5 × 5 × 5 × 5
c 7 × 7 × 7 × 7 × 7
d 6 × 6 × 6
e 3 × 3 × 3 × 3 × 3 × 3
f n × n × n × n × n × n × n
g a × a × a × a
h k × k × k × k × k × k × k × k × k × k
4 a 243
b 256
c 256
d 1331
e 2401
f 216
g 1
h 625
5 a B
b D
6 a 22 × 44 × 6
b 34 × 74
c 23 × 195
d 44 × 132
5 2 2
e 66m p s f 378i3n2r3
g 192e3kp2
h 99j5lp2
7 a 15 × f × f × f × j × j × j × j
b 7 × k × k × k × k × k × k × s × s
c 4 × b × b × b × c × c × c × c × c
d 19 × a × a × a × a × m × n × n × n
e 8 × l × l × l × l × r × r × t × t
8 a 64 = 26
b 40 = 23 × 5
d 400 = 24 × 52
c 36 = 22 × 32
2
2
e 225 = 3 × 5 f 2000 = 24 × 53
9 a 120
b 100
d 308
c 216
e 1575
f 760 000
10 a i 101
ii 102
iii 103
iv 106
2
b ii 5 × 10
iii 4 × 102 + 7 × 101
iv 2 × 103 + 3 × 102 + 6 × 101
v 1 × 103 + 9 × 102 + 8 × 101
vi 5 × 103 + 4 × 102 + 3 × 101
c i 75 000
ii 30 600
iii 5 200 480
11 Factors multiplied together in ‘shorthand’ form:
a×a×a×b×b×b×b
1
b 2 074 464
2 a23a2b3c4
1
3 a1 × 102 + 4 × 101, 6 × 102 + 8 × 101
b7 × 102 + 12 × 101 = 8 × 102 + 2 × 101
c820
d820
e Check with your teacher.
14 a1 × 23 + 1 × 21, 1 × 23 + 1 × 22 + 1 × 21
b2 × 23 + 1 × 22 + 2 × 21 = 1 × 24 + 1 × 22 + 1 × 22
= 1 × 24 + 2 × 22 = 1 × 24 + 1 × 23
c24
d24
e Check with your teacher.
Challenge 3.1
312 minutes
3.4 Second Index Law (dividing numbers in index form with
the same base)
1 a
2 a
e
i
3 a
e
23
3
33
1533
3x2
8f 9
i 8g3
4 a D
5 a 3p4
e 20r4
3p5
2
74
117
13
h77
6y2
10h90
3b6
j
2
b
b
f
j
b
f
b 6r4
f 9q
8b5
3
4p2rs
f 3f 3j5
g
3
10
−
p
4e
+
4
7 a 2
b 2
23 × 24 × 22
6 a
b
103
54
64
b70
8w7
4j10
9m4
k
2
b A
c 9a3
c
c
g
k
c
g
125
104
f 900
3q4
5p10
5n90
l
2
d
h
l
d
h
d 3b6
9x8y
4hk3
5m10n6
d
e
3
4
6
20f 6g2h4
9a5b3c
h
i
2
3
3x
+
y
c 5
d 31 − m
c
8 a
21 × 25
b i 23
ii 8
c 8
2 × 3 × 33 × 22 × 32
9 a
b i 21
22 × 3 × 34
× 31
ii 6
64 Maths Quest 8
c03IndexLaws.indd 64
09/05/16 11:02 AM
number and algebra
7 a 2g8
f 15a10
not apply.
11 3p− q
12 ap ÷ ap = 1 and ap ÷ ap = ap− p = a0 ⇒ a0 = 1
1
6
m8
=
1
m
2
1
⇒ m −2 =
m2
Challenge 3.2
54 dabs
3.5 Third Index Law (the power of zero)
b 1
b 1
b 12
3
d 1
d 1
d 22
g 3p2
c 1
8
d
7
h 1
d 7
c 2
d 2p9
f y
g
h
b A
b 2
c
c
4 a 12m b 21
e 2
5 a 1
e 3
6 a 1
f
b
f
b
3
e e4
2
7 a D
8 a 1
c 1
c 1
c −9
10b2
1
3
4
c 72
m3
3
D
2
q4
5
c3
4
c 11100
330
f 135 × 1715
g
2 a p8 q6
b r15w9
c b10n18
d j18g12
g f 16a21
3 a 220
f g39
4 a B
5 a a6
e f 17
i c20
9b8
6 a
d6
125y21
e
h
b
g
b
b
f
j
q4r20
t10u8
t33
324a20
B
m4
g6
f 7
25h20
b
4j4
256a12
f
i
c
h
c
c
g
k
Standard
form
d 2
27z39
f
2401c20
d 5144
e 38 × 1012
h 34w36q8 i
E
h24j16
i15j12
a21
216d27
D
n3
p9
k14
8k15
c
27t24
M
PL
SA
e
220
72e10
r4q8
d b24
e e66
i 40 000r54
d b8
h y2
l p16
d
o 64p6q15
b 6x2
Investigation — Rich task
EV
b 680
n xy
2
3.6 Fourth Index Law (raising a power to another power)
1 a 36
m 15a9b5
c 12a11
d 8x2
e 7d3 + d2
2
f k2
g h 6s7t5
i bc4
j 0
3
k 18p
l 16x9y7 m 8a6b2
n xy
o 6q2
9 Any base, even a complex one like this, raised to the power of
one equals zero.
10 Karla has added the powers instead of multiplying.
r7 ÷ r6 = r1 = r
The correct answer is r6.
(2 )
4
11 2(2 ) = 22 = 216; (((2) 2) 2) 2 = 28
5
3
2
2
2
12 22 ; 23 ; 25 ; 22 ; 32 ; 52
8
12
1
4
13 a (16x y ) , (4x y6) 2, (2x2y3) 4
b (312na6nb12n) 1, (312a6b12) n, (36a3b6) 2n, (32a1b2) 6n,
(36na3nb6n) 2, (32nanb2n) 6
49p18
64q44
Basic numeral
Name
SI
prefix
SI
symbol
1.0 × 1012
1 000 000 000 000
Trillion
tera
T
1.0 × 10
1 000 000 000
Billion
giga
G
1.0 × 106
1 000 000
Million
mega
M
1.0 × 10
1000
Thousand
kilo
k
1.0 × 102
100
Hundred
hecto
h
1.0 × 101
10
Ten
deca
da
1.0 × 10−1
0.1
Tenth
deci
d
1.0 × 10−2
0.01
Hundredth
centi
c
1.0 × 10
0.001
Thousandth
milli
m
1.0 × 10−6
0.000 001
Millionth
micro
m
1.0 × 10−9
0.000 000 001
Billionth
nano
n
1.0 × 10−12
0.000 000 000 001
Trillionth
pico
p
9
AL
U
f
l x3y7
2
h2
n3
g
h
2
5
i v2
j 2x6
9 Any base raised to the power of one equals zero.
20x2 x2 x2
10 20 = 1, x0 = 1 ⇒
=
=
22x0 22 4
1
11
3
12 Peter has treated the x’s incorrectly. He has calculated x6 ÷ 2
instead of x6 − 2.
The answer is 4x4.
If 64 and 16 are converted to base 2 or base 4, and the indices
divided not subtracted, an error will occur.
e 2
e 4d 6
j f 8g6
O
1 a 1
2 a 1
3 a 4
d 6x5
i 2x4
N
14
m6
c w18
h 60b4c8
AT
IO
13 m6 ÷ m8 = m6− 8 = m− 2 but m6 ÷ m8 =
8u3v4
3
8 a w10
k
b 8p8
g 3s6
N
LY
10 The bases, 12 and 3, are different. The laws of indices do
3
−3
1 3.0 × 105 km/s
2 1.8 × 107 km/min
3 1.08 × 109 km/h
4 2.592 × 1010 km/day
5 9.47 × 1012 km. On average, there are 365.25 days in one year.
6 4.07 × 1013 km
7 Check with your teacher.
Code puzzle
The liver produces a litre of bile a day. It helps break down fat.
Topic 3 • Index laws 65
c03IndexLaws.indd 65
09/05/16 10:05 AM
ICT activity
N
LY
Task
Watch the case study video that ACSEUP has provided
for your class, and simultaneously track the growth of
the killer balloons on a chart. Use the data that you have
collected to create a graph showing the growth rate of
the killer balloons. From these, create a mathematical
equation that simulates the attack.
When you have finished this research and understand
the problem, you will answer some other questions that
ACSEUP has asked about similar balloon scenarios.
These will include how to go about defeating the killer
balloons. Using your solutions to these questions,
combined with your charts, graphs and equations, you
will present a comprehensive report about the attack of
the killer balloons to ACSEUP.
SA
M
PL
E
EV
AL
U
Ms Lovely is a Math teacher at Scholar High School and
during last week she had her 30th birthday.
Mr Handsome, her adoring
husband, secretly placed a
balloon and flowers in her
classroom before the
beginning of her first class.
Unfortunately, he had
unknowingly purchased a killer
balloon from Mr Loon, of Angry
Balloons! During Ms Lovely’s class,
the balloon developed a strangelooking face and burst open.
To everyone’s amazement, there
were now five balloons, all with
strange faces!
During the following period when
Ms Lovely was taking a new class, the five balloons also
burst, and during the day they continued to multiply in this
O
Scenario
N
Searchlight ID: pro-0095
AT
IO
Attack of the killer
balloons
way. Eventually it became apparent that the balloons were
hostile and were going to attack.
Terrified, Ms Lovely and the threatened children
escaped, but the killer balloons had taken control of
her classroom and were threatening to spread. The
students and staff were evacuated from the school,
and the Australian Centre for the Study of Extraordinary
and Unexplained Phenomena (ACSEUP) was called to
investigate. ACSEUP re-created the events to produce
a simulation video so they could study the balloons and
work out how to defeat them. They realised that if they
could work out the mathematical pattern of the increasing
numbers of balloons perhaps they could find a plan for
defeating them. However, after extensive analysis, they
were stumped. The ACSEUP researchers had heard
that your school had some clever maths students, and
have asked your mathematics class to help Ms Lovely’s
students solve the mystery of the killer balloons.
66
Maths Quest 8
c03IndexLaws.indd 66
09/05/16 2:23 PM
•
•
Your ProjectsPLUS application is available in
this topic’s Student Resources tab inside your
eBookPLUS. Visit www.jacplus.com.au to locate
your digital resources.
EV
Interactivity
N
LY
•
AL
U
• Open the ProjectsPLUS application for this topic
in your eBookPLUS. Watch the Attack of the Killer
Balloons case study, navigate to your Media Centre,
and print Template 1 provided in the Template section.
Fill in the required data in Template 1. Your Media
Centre includes the re-enactment of the killer balloon
take-over.
• Next, press the ‘Start Project’ button and then set up
your project group. You will need to create a group of
two or three of your classmates before you begin this
•
O
process
•
AT
IO
• Record and graph the growth of the balloons.
• Create a mathematical formula that simulates the
growth of the balloons.
• Answer questions about other balloon growth
scenarios.
• Record and graph the popping of the balloons.
• Create a mathematical formula that simulates the
destruction of the balloons.
• Answer questions about other balloon destruction
scenarios.
• Prepare a report on the attack of the killer balloons for
ACSEUP.
project. Save your group
SuggeSTed
settings and the project will
be launched.
SoFTWare
Navigate to the Media
• ProjectsPLUS
Centre, then to the
• Microsoft Word
Documents section and
• GeoGebra
answer the questions in the
six ACSEUP worksheets.
Navigate to the Media Centre, then to the Template
section and print Presentation guidelines.
Use the Presentation guidelines to choose a topic for
your presentation, then read about the guidelines for
the project.
Navigate to the Media Centre to access GeoGebra and
the corresponding user manual.
If possible, use a SMARTBoard or overhead projector
and computer to display your GeoGebra graph and
tables during the presentation.
N
Summary of tasks
SA
M
PL
E
KILLER BALLOONS
SearCHlIgHT Id: int-2447
If a quantity doubles or triples over a consistent
time period, it is said to be increasing
exponentially. Use this interactivity to change
the exponential rate of growth and then predict
the number of balloons that will appear. Watch
the graph of the number of balloons against time
plotted on the wall.
MEDIA CENTRE
Your Media Centre contains:
• a Document section with
ACSEUP worksheets
• a Template section with
presentation guidelines
• GeoGebra software and its
user manual
• an assessment rubric.
Topic 3 • The
ICTlargest
activity
gland
— projectsplus
in the body 67
c03IndexLaws.indd 67
09/05/16 10:06 AM