NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
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The first cubic number, 1, equals 1 ì 1 ì 1.
The second cubic number, 8, equals 2 ì 2 ì 2.
The third square number, 27, equals 3 ì 3 ì 3.
If this pattern is continued, any cubic number can be found by writing the position of the
cubic number 3 times and multiplying. This is known as cubing a number and is written by an
index (or power) of 3. For example, 43 = 4 ì 4 ì 4 = 64.
WORKED EXAMPLE 14
Find the value of 53.
THINK
WRITE
1
Write 53 as the product of three lots of 5.
2
Evaluate.
53 = 5 ì 5 ì 5
= 125
Cube roots
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The opposite of cubing a number is finding the cube root of a number.
The cube root is found by looking for a number that can be written three times and multiplied
to produce the given number.
The cube root symbol is similar to the square root symbol but with a small 3 written in front,
as shown: 3 .
From the worked example, we can see that 3 125 = 5.
WORKED EXAMPLE 15
Find 3 27 .
THINK
WRITE
27 = 3 ì 3 ì 3
Look for a number that when written three times and multiplied
gives 27.
3
27 = 3
REMEMBER
1. When a number is written three times and multiplied, this is called cubing a number
and is written using an index (or power) of 3.
2. The opposite of cubing a number is finding the cube root of a number. This is
calculated using our knowledge of cubic numbers or using a calculator.
EXERCISE
3D
Cubes and cube roots
FLUENCY
1 WE14 Find the value of 43.
2 Find the value of:
a 23
b 33
3 Write the first 10 cubic numbers.
4 WE15 Find 3 8 .
82
Maths Quest 7 for the Australian Curriculum
c 63
d 103
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
INDIVIDUAL
PATHWAYS
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Activity 3-D-1
Cubes and cube
roots
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Activity 3-D-2
More cubes and
cube roots
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Activity 3-D-3
Advanced cubes and
cube roots
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UNDERSTANDING
5 Find each of the following. Verify your answers with a calculator.
a 3 64
b
3
216
c
3
343
3
d
729
(Hint: Use your answer to question 3.)
REASONING
6 The first 5 square numbers are 1, 4, 9, 16, 25. If we find the difference between these numbers,
we get 4 – 1 = 3, 9 – 4 = 5, 16 – 9 = 7 and 25 – 16 = 9. These numbers all differ by 2.
Representing this in a table, we get:
Square numbers
First difference
Second difference
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1
4
3
9
5
2
16
7
2
25
9
2
Repeat this process for the first 6 cubic numbers. How many times did you need to find the
difference until they were equal?
If you look at 14, 24, 34, 44, . . ., how many differences would you need to find until it they
were equal?
REFLECTION
What would be the first 4
numbers that could be arranged
as a triangle-based pyramid (all
triangles equilateral)?
Chapter 3
Indices and primes
83
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
Summary
Index notation
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Index notation is a shorthand way of writing a repeated multiplication.
The number being multiplied is the base of the expression; the number of times it has been
written is the index (or power).
Prime numbers and composite numbers
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A prime number is a whole number which has exactly two factors, itself and 1.
20
A composite number is one which has more than two factors.
The number 1 is neither a prime number nor a composite number.
A factor tree shows the prime factors of a composite number.
4
The last numbers in the factor tree are all prime numbers, therefore they
are prime factors of the original number.
2
2
Every composite number can be written as a product of prime factors; for
example, 20 = 2 ì 2 ì 5.
The product of prime factors can be written in shorter form using index notation.
Lowest common multiples and highest common factors can be determined by expressing
numbers as the products of their prime factors.
5
Squares and square roots
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Square numbers are written using an index (or power) of 2 and are found by multiplying the
number by itself.
The opposite of squaring a number is finding the square root of a number.
To find the square root of a number we can use our knowledge of square numbers or use a
calculator.
Cubes and cube roots
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■
When a number is written three times and multiplied, this is called cubing a number and is
written using an index (or power) of 3.
The opposite of cubing a number is finding the cube root of a number. This is calculated using
our knowledge of cubic numbers or using a calculator.
MAPPING YOUR UNDERSTANDING
Homework
Book
84
Using terms from the summary above, and other terms if you wish, construct a concept map
that illustrates your understanding of the key concepts covered in this chapter. Compare
your concept map with the one that you created in What do you know? on this chapter’s
opening page. Have you completed the two Homework sheets, the Rich task and two Code
puzzles in your Maths Quest 7 Homework book?
Maths Quest 7 for the Australian Curriculum