Download Index notation

NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
Index notation
3A
■
■
An index is a shorthand way of writing a repeated multiplication.
The base is the number that is being multiplied and the index (or power) is the number of
times it is multiplied.
index (or power)
2 ì 2 ì 2 ì 2 = 24
base
WORKED EXAMPLE 1
Write the following using index notation.
a 5ì5ì5ì5ì5ì5ì5
b 3ì3ì3ì3ì7ì7
THINK
a
b
WRITE
1
Write the multiplication.
2
Write the number being multiplied as the base
and the number of times it is written as the index.
1
Write the multiplication.
2
Write the number being multiplied as the base
and the number of times it is written as the index.
■
a
5ì5ì5ì5ì5ì5ì5
= 57
b
3ì3ì3ì3ì7ì7
= 34 ì 72
By using place value, you can write numbers in expanded form with index notation.
2700 = 2000 + 700 = 2 ì 10 ì 10 ì 10 + 7 ì 10 ì 10 = 2 ì 103 + 7 ì 102
WORKED EXAMPLE 2
Write the following numbers in expanded form using index notation.
a 59 176
b 108 009
THINK
a
b
WRITE
a
1
Read the number to yourself, stating the place
values.
2
Write the number as the sum of each place value.
59 176 = 50 000 + 9000 + 100 + 70 + 6
3
Write each place value in multiples of 10.
59 176 = 5 ì 10 ì 10 ì 10 ì 10 + 9 ì 10
ì 10 ì 10 + 1 ì 10 ì 10 + 7
ì 10 + 6
4
Write each place value in index notation.
59 176 = 5 ì 104 + 9 ì 103 + 1 ì 102 + 7
ì 101 + 6
1
Read the number to yourself stating the place
values.
2
Write the number as the sum of each place value.
108 009 = 100 000 + 8000 + 9
3
Write each place value in multiples of 10.
108 009 = 1 ì 10 ì 10 ì 10 ì 10 ì 10
+ 8 ì 10 ì 10 ì 10 + 9
4
Write each place value in index notation.
108 009 = 1 ì 105 + 8 ì 103 + 9
b
Chapter 3
Indices and primes
69
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
WORKED EXAMPLE 3
Simplify each of the following by first writing as a repeated multiplication and then using index
notation.
a 34 ì 36
b (43)3
c (3 ì 5)2
THINK
a
b
c
WRITE
a
34 ì 36
1
Write the question.
2
Write the question using repeated multiplication
(that is, in expanded form).
= (3 ì 3 ì 3 ì 3) ì (3 ì 3 ì 3 ì 3 ì 3 ì 3)
=3ì3ì3ì3ì3ì3ì3ì3ì3ì3
3
Write the repeated multiplication using index notation.
Note: 34 ì 36 = 310.
= 310
1
Write the question.
2
Write the question using repeated multiplication
(that is, in expanded form).
= 43 ì 43 ì 43
= (4 ì 4 ì 4) ì (4 ì 4 ì 4) ì (4 ì 4 ì 4)
=4ì4ì4ì4ì4ì4ì4ì4ì4
3
Write the repeated multiplication using index notation.
Note: (43)3 = 49.
= 49
1
Write the question.
2
Write the question using repeated multiplication
(that is, in expanded form).
= (3 ì 5) ì (3 ì 5)
=3ì5ì3ì5
=3ì3ì5ì5
3
Write the repeated multiplication using index notation.
Note: (3 ì 5)2 = 32 ì 52.
= 32 ì 52
b
c
(43)3
(3 ì 5)2
REMEMBER
1. Index notation is a shorthand way of writing a repeated multiplication.
2. The number being multiplied is the base of the expression; the number of times it has
been written is the index (or power).
EXERCISE
3A
INDIVIDUAL
PATHWAYS
eBoo
k plus
eBook
Activity 3-A-1
Famous astronomer
doc-1675
Activity 3-A-2
Famous
mathematician
doc-1676
Activity 3-A-3
Famous physicist
doc-1677
70
Index notation
FLUENCY
1 WE1a Write 7 ì 7 ì 7 ì 7 using index notation.
2 Write each of the following using index notation.
a 2ì2ì2ì2
b 8ì8ì8ì8ì8ì8
c 10 ì 10 ì 10 ì 10 ì 10
d 3ì3ì3ì3ì3ì3ì3ì3ì3
e 6ì6ì6ì6ì6ì6ì6
f 13 ì 13 ì 13
g 12 ì 12 ì 12 ì 12 ì 12 ì 12
h 9ì9ì9ì9ì9ì9ì9ì9
3 WE1b Write 4 ì 4 ì 4 ì 4 ì 4 ì 6 ì 6 ì 6 using index notation.
Maths Quest 7 for the Australian Curriculum