number AND algebra • Number and place value
3A
Review of index form
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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 ì 2 ì 2 ì 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
Index form
= 2 ì 2 ì 2 ì 2 =
Factor form
16
Basic numeral
Worked Example 1
State the base and power for the number 514.
Think
Write
1
Write the number.
514
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.
Worked Example 2
Write 124 in factor form.
Think
1
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.
Write
124
= 12 ì 12 ì 12 ì 12
Worked Example 3
Write 2 ì 5 ì 2 ì 2 ì 5 ì 2 ì 5 in index form.
Think
Write
1
Write the problem.
2ì5ì2ì2ì5ì2ì5
2
Write the factors in numerical order.
=2ì2ì2ì2ì5ì5ì5
3
The number 2 has been written 4 times and
multiplied. The number 5 has been written 3
times and multiplied.
= 24 ì 53
Chapter 3 Index laws
51
number AND algebra • Number and place value
Worked Example 4
Write 7 ì 53 ì 65 in factor form.
Think
Write
7 ì 53 ì 65
1
Write the problem.
2
List the factors: 7 is written once, 5 is written 3 times
=7ì5ì5ì5ì6ì6ì6ì6ì6
and multiplied, and 6 is written 5 times and multiplied.
remember
1. A number written in index form has:
(a) a base, which is the large number below
(b)a power or index, which is the small number written just above and to the right of
the base.
base ç 25 ã index
2. The base tells us what will be multiplied.
3. The power tells us how many times the base will be written and multiplied.
4. Factor form is when all the multiplications are shown.
5. After multiplying out to remove the index, the answer is called a basic numeral.
index form
Exercise
3a
Individual
Pathways
eBook plus
Activity 3-A-1
Review of index form
doc-2143
Activity 3-A-2
More index form
doc-2144
Activity 3-A-3
Advanced index form
doc-2145
factor form
basic numeral
Review of index form
Fluency
1 WE 1 State the base and power for each of the following.
a 84
b 710
c 2011
d 190
100
5
24
f 3
g m
h c
i n36
2 Write the following in index form.
a 2 ì 2 ì 2 ì 2 ì 2 ì 2
c x ì x ì x ì x ì x
e 11 ì l ì l ì l ì l ì l ì l ì l
e 7812
j d42
b 4 ì 4 ì 4 ì 4
d 9 ì 9 ì 9
f 44 ì m ì m ì m ì m ì m
3 WE 2 Write the following in factor form.
a 42
b 54
c 75
f n7
g a4
h k10
4 Write each of the following as a basic numeral.
a 35
b 44
c 28
f 63
g 110
h 54
52
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25 = 2 ì 2 ì 2 ì 2 ì 2 = 32
d 63
d 113
e 36
e 74
5 a MC What does 63 mean?
a 6 ì 3
d 6 + 6 + 6
b 6 ì 6 ì 6
E 3 ì 6
c 3 ì 3 ì 3 ì 3 ì 3 ì 3
b What does 35 mean?
a 3 ì 5
d 3 ì 3 ì 3 ì 3 ì 3
b 5 ì 5
E 5 ì 3
c 3 + 3 + 3 + 3 + 3
Maths Quest 8 for the Australian Curriculum