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Table of Contents
Squares, Square Roots & Perfect Squares
Numerical
Roots and Radicals
Square Roots of Numbers Greater than 400
Estimating Square Roots
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Area of a Square
The area of a figure is the number of square units needed to cover
the figure.
Squares, Square Roots
and Perfect Squares
The area of the square below is 16 square units because 16
square units are needed to COVER the figure...
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Table of
Contents
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Area of a Square
The area (A) of a square can be found by squaring its
side length, as shown below:
2
A=s
A = s2
A = 42 = 4 4 = 16 sq.units
4 units
The area (A) of a square is
labeled as square units, or
units2, because you cover the
figure with squares...
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1
What is the area of a square with sides of 5 inches?
A
16 in2
B
20 in2
C
25 in2
D
30 in2
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2
What is the area of a square with sides of 6 inches?
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3
If a square has an area of 9 ft2, what is the length of a side?
A
16 in2
A
2 ft
B
20 in2
B
2.25 ft
C
2
24 in
C
3 ft
D
36 in2
D
4.5 ft
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When you square a number you multiply it by itself.
52 = 5 5 = 25 so the square of 5 is 25.
You can indicate squaring a number with an exponent of 2,
by asking for the square of a number, or by asking for a
number squared.
What is the square of seven?
What is nine squared?
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Make a list of the numbers 1-15 and then square each of
them.
Your paper should be set up as follows:
Number
1
2
3
(and so on)
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Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
The numbers in the right column
are squares of the numbers in the
left column.
If you want to "undo" squaring a
number, you must take the square
root of the number.
So, the numbers in the left column
are the square roots of the
numbers in the right column.
Square
1
4
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Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
The square root of a number is found by
undoing the squaring. The symbol for
square root is called a radical sign and
it looks like this:
Using our list, to find the square root of
a number, you find the number in the
right hand column and look to the left.
So, the 81 = 9
What is 169?
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Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
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Find the following.
You may refer to your chart if you need to.
When the square root of a number is a
whole number, the number is called a
perfect square.
Since all of the numbers in the right
hand column have whole numbers for
their square roots, this is a list of the
first 15 perfect squares.
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4
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5
What is 1 ?
What is 81 ?
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6
What is the square of 15?
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7
What is 256 ?
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8
What is 132?
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9
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10 What is the square of 18?
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12 What is 20 squared?
What is 196 ?
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11 What is 11 squared?
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13 What is the square root of 400?
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Think about this...
What about larger numbers?
How do you find
Square Roots of Numbers
Greater than 400
?
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Table of
Contents
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It helps to know the squares of larger numbers such as the
multiples of tens.
For larger numbers, determine between which
two multiples of ten the number lies.
102 = 100
202 = 400
302 = 900
402 = 1600
502 = 2500
602 = 3600
702 = 4900
802 = 6400
902 = 8100
1002 = 10000
102 = 100
202 = 400
302 = 900
402 = 1600
502 = 2500
602 = 3600
702 = 4900
802 = 6400
902 = 8100
1002 = 10000
What pattern do you notice?
Next, look at the ones digit to determine the ones
digit of your square root.
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Lies between 6400 and 8100 (80 and 90)
Ends in 4 so square root ends in 2 or 8
Try 82 then 88
822 = 6724 NO!
882 = 7744
=
=
=
=
=
=
=
=
=
=
1
4
9
16
25
36
49
64
81
100
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Examples:
List of
Squares
14 Find.
List of
Squares
Lies between 2500 & 3600 (50 and 60)
Ends in nine so square root ends in 3 or 7
Try 53 then 57
532 = 2809
12
22
32
42
52
62
72
82
92
102
15 Find.
17 Find.
19 Find.
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16 Find.
18 Find.
20 Find.
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List of
Squares
List of
Squares
List of
Squares
List of
Squares
List of
Squares
List of
Squares
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List of
Squares
22 Find.
List of
Squares
21 Find.
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List of
Squares
23 Find.
Estimating Square Roots
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Table of
Contents
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All of the examples so far have
been from perfect squares.
What does it mean to be a perfect square?
The square of an integer is a perfect square.
A perfect square has a whole number square root.
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You know how to find the square root of a perfect square.
What happens if the number is not a perfect square?
Does it have a square root?
What would the square root look like?
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Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Think about the square root of 50.
Where would it be on this chart?
What can you say about the
square root of 50?
50 is between the perfect squares
49 and 64 but closer to 49.
So the square root of 50 is
between 7 and 8 but closer to 7.
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Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
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Square
Root
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Estimate the following:
200
215
24 The square root of 40 falls between which two perfect
squares?
3 and 4
B
5 and 6
C
6 and 7
D
7 and 8
· Between which two perfect squares it
lies (and therefore which 2 square roots).
· Which perfect square it is closer to (and
therefore which square root).
Example: 110
Lies between 100 & 121, closer to 100.
So 110 is between 10 & 11, closer to 10.
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Another way to think about it is to use a number line.
30
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A
When estimating square roots of
numbers, you need to determine:
#4
#8 #9
2
3
Since 8 is closer to 9 than to 4, #8 is closer to 3 than
to 2, so #8 # 2.8
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25 Which whole number is 40 closest to?
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26 The square root of 110 falls between which two perfect
squares?
A
6 and 7
B
7 and 8
C
8 and 9
D
10 and 11
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27 Estimate to the nearest whole number.
110
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28 Estimate to the nearest whole number.
29 Estimate to the nearest whole number.
213
90
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