Download Squares and square roots

NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
3C
Squares and square roots
Square numbers
■
Square numbers are numbers that can be arranged in a square, as shown below.
12 = 1
22 = 4
32 = 9
42 = 16
■
■
■
■
■
The first square number, 1, equals 1 ì 1.
The second square number, 4, equals 2 ì 2.
The third square number, 9, equals 3 ì 3.
The fourth square number, 16, equals 4 ì 4.
If this pattern is continued, any square number can be found by multiplying the position of the
square number by itself. This is known as squaring a number and is written by an index (or
power) of 2. For example, 3 ì 3 can be written as 32.
WORKED EXAMPLE 10
Find the square numbers between 90 and 150.
THINK
1
Use your knowledge of tables to find the first square number
after 90.
102 = 10 ì 10 = 100
2
Find the square numbers which come after that one but
before 150.
112 = 11 ì 11 = 121
122 = 12 ì 12 = 144
132 = 13 ì 13 = 169
3
78
WRITE
Write the answer in a sentence.
Maths Quest 7 for the Australian Curriculum
(too big)
The square numbers between 90 and
150 are 100, 121 and 144.
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
WORKED EXAMPLE 11
Write the two whole numbers between which 5.72 will lie.
THINK
WRITE
1
Write the whole numbers either side of 5.7.
5.7 is between 5 and 6.
2
Consider the square of each whole number.
5.72 is between 52 and 62.
3
Simplify 52 and 62 then write the answer in a sentence.
So 5.72 is between 25 and 36.
4
Verify your answer with a calculator.
Square roots
■
■
■
■
Evaluating the square root of a number is the opposite of squaring a number.
By writing 42 = 16, we mean that 4 multiplied by itself is equal to 16. The opposite of this is
to say that the square root of 16 is 4, or 16 = 4.
Using the illustrations of the squares above, a square of area 16 square units must have a side
length of 4 units.
To find the square roots of larger numbers, it helps to break the number up as a product of
two smaller square roots with which we are more familiar. For example,
900 = 9 ì 100
= 3 ì 10
= 30
WORKED EXAMPLE 12
Find: a
49
b
3600 .
THINK
a
b
Find a number which when multiplied by itself gives
49 (7 ì 7 = 49).
WRITE
a
b
49 = 7
1
Write 3600 as the product of two smaller
numbers for which we can find the square root.
2
Take the square root of each of these numbers.
= 6 ì 10
3
Find the product.
= 60
3600 =
36 ì 100
WORKED EXAMPLE 13
Between which two numbers will 74 lie?
THINK
1
Write the square numbers either side of 74.
2
Consider the square root of each number.
3
Simplify 64 and 81.
4
Verify your answer with a calculator.
WRITE
74 is between 64 and 81.
74 is between 64 and 81.
So 74 is between 8 and 9.
Chapter 3
Indices and primes
79
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
REMEMBER
1. Square numbers are written using an index (or power) of 2 and are found by
multiplying the number by itself.
2. The opposite of squaring a number is finding the square root of a number.
3. To find the square root of a number we can use our knowledge of square numbers
or use a calculator.
EXERCISE
3C
INDIVIDUAL
PATHWAYS
eBoo
k plus
eBook
Activity 3-C-1
Squares and square
roots
doc-1681
Activity 3-C-2
More squares and
square roots
doc-1682
Activity 3-C-3
Advanced squares
and square roots
doc-1683
Squares and square roots
FLUENCY
1 Evaluate and verify your answers with a calculator.
a 82
b 112
c 152
d 252
2 WE10 a Find the square numbers between 50 and 100.
b Find the square numbers between 160 and 200.
3 WE11 Write two whole square numbers between which each of the following will lie.
a 6.42
b 7.82
c 9.22
d 12.52
4 WE12a Find:
a
25
81
b
c
144
d
400
d
160 000
5 WE12b Find and verify your answers with a calculator.
a
4900
b
14 400
c
360 000
6 WE13 Between which two numbers will 60 lie?
7 Write the two whole numbers between which each of the following will lie.
a
14
b
90
c
200
2
d
UNDERSTANDING
8 a Find the even square numbers between 10 and 70.
b Find the odd square numbers between 50 and 120.
9 MC For which of the following square roots can we calculate an exact answer?
A
10
B
25
C
50
D
75
E
82
10 MC For which of the following square roots can we NOT calculate the exact value?
A
160
B
400
C
900
D
11 Evaluate the following. Verify your answers with a calculator.
a 22 +
25
b 92 - 36
c 52 ì 22 ì
49
d 32 + 22 ì 16
e 32 - 22 ó
f
80
4 + 49
9 ì 42 - 144 ó 22
Maths Quest 7 for the Australian Curriculum
2500
E
3600