Download Index notation

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
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Activity 3-A-2
Famous
mathematician
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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
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
4 Write the following using index notation.
a 2ì2ì3
c 5ì5ì2ì2ì2ì2
e 5 ì 11 ì 11 ì 3 ì 3 ì 3
g 2ì2ì2ì3ì3ì5
3ì3ì3ì3ì2ì2
7ì2ì2ì2
13 ì 5 ì 5 ì 5 ì 7 ì 7
3ì3ì2ì2ì5ì5ì5
5 Write 65 using a repeated multiplication (that is, in expanded form).
6 Write each of the following using repeated multiplication (that is, in expanded form).
a 113
b 49
c 56
7 WE2 Write the following numbers in expanded form using index notation.
a 300
b 4500
c 6785
d 10 000
e 31 702
f 200 301
b
d
f
h
8 WE3 Simplify each of the following by first writing as a repeated multiplication and then
using index notation.
a 58 ì 53
d (27)3
g (3 ì 13)6
b 72 ì 76
e (113)8
h (2 ì 5)4
c 48 ì 45
f (95)3
i (5 ì 7)5
UNDERSTANDING
9 a MC Which of the following expressions has the greatest value?
A 28
B 82
C 34
D 43
E 92
b Justify your answer using mathematical reasoning.
10 MC The value of 44 is:
A 8
B 16
11 Find the value of
23
ì
C 64
D 256
E 484
32.
12 Evaluate each of the following.
a 34 ì 43
b 35 + 93
4
3
d 6 -9
e 53 + 25 ì 92
c 83 ó 25
f 27 - 45 ó 26
REASONING
13 We know that 122 = 144 and 212 = 441. It is also true that 132 = 169 and 312 = 961.
If 142 = 196 will 412 = 691? Try to justify your answer without calculating 412.
14 There is a shortcut that allows you to square any
number that ends in 5. Look at the following examples
REFLECTION
and see if you can identify the rule. Use your rule to
evaluate 752 and 952.
Why do we use index notation?
152 = 225, 252 = 625, 352 = 1225, 452 = 2025
3B
Prime numbers and composite
numbers
Prime numbers
■
■
■
■
■
■
A prime number is a counting number that has exactly 2 factors, itself and 1.
(Counting numbers are 1, 2, 3, 4, . . . .)
The number 3 is a prime number. Its only factors are 3 and 1.
The number 2 is the only even prime number. Its only factors are 2 and 1.
A composite number is one which has more than two factors.
The number 4 is composite. Its factors are 1, 2 and 4.
The number 1 is a special number. It is neither a prime number nor a composite number
because it has only one factor, 1.
Chapter 3
Indices and primes
71