Download M-30 12 pg 4 add sub sqrts.cwk (WP)

Chapter 12 Square Roots
Perfect Squares and Square Roots
What numbers are Perfect Squares?
When a positive number is multiplied by itself the number obtained is called a Perfect Square.
1•1=
2 •2 =
1
3• 3 =
4
4 • 4 = 16
5 • 5 = 25
9
6 • 6 = 36
7• 7 = 49
8 • 8 = 64
9 • 9 = 81
10 •10 = 100
11•11= 121
12 •12 = 144
13•13 = 169
14 •14 = 196
15 •15 = 225
Not all positive whole numbers are perfect squares.
30 is not a perfect square because there is not a positive number times itself that equals 30. 45 is
also not a perfect square because there is no positive number times itself that equals 45. The
numbers in the squares above are the only whole numbers from 1 to 225 that are perfect squares.
All the other whole numbers from 1 to 225 are not perfect squares.
Some positive fractions are perfect squares. When a positive fraction is multiplied by itself the
fraction obtained is called a Perfect Square.
2 2
• =
3 3
4
9
12 12
144
• =
11 11
121
5 5
• =
4 4
25
16
3 3
9
• =
7 7
49
9 9
• =
8 8
81
64
6 6
36
• =
13 13
169
1 1
1
• =
10 10
100
14 14
196
• =
15 15
225
Some fractions would be perfect squares if they were reduced.
8
8 4
does not seem to be a perfect square. When it is reduced
=
is easy to see that the
18
18 9
32
32
reduced form is a perfect square. The reduced form of
is also a perfect square because
50
50
16
reduces to
which is a perfect square.
25
Chapter 12
© 2015 Eitel
Finding the Square Root of a Perfect Square
The symbol
is called a radical sign but it is also commonly called a square root sign. The
square root of a number is written as a where a is a positive number.
a is read as "the square root of a"
9 is read as "the square root of 9
25 is read as "the square root of 25
a asks you "What positive number times itself is equal to a?
36 asks "What positive number times itself equals 36. Since 6 • 6 = 36 then
36 = 6
100 asks "What positive number times itself equals 100. Since 10 •10 = 100 then
100 = 10
16
16
4 4 16
asks "What positive number times itself equals
Since
• =
then
81
81
9 9 81
16 4
=
81 9
Example 1
Find
Find
9
3• 3 = 9 so
9 =3
25
5 • 5 = 25 so
25 = 5
Example 4
Find
Example 2
196
Example 5
Find
4
Example 3
Find
81
9 • 9 = 81 so
81 = 9
Example 6
Find
144
14 • 14 = 196 so
2 • 2 = 4 so
12 •12 = 144 so
196 = 14
4 =2
144 = 12
Example 7
Find
36
49
Example 8
Find
100
121
6 6 36
• =
so
7 7 49
10 10 100
• =
so
11 11 121
36 6
=
49 7
100 10
=
121 11
Chapter 12
Example 9
Find
18
50
18
9
=
50
25
9
3
=
25 5
© 2015 Eitel
Square Roots
Find the following square roots.
1.
9
2.
25
3.
81
4.
5.
100
6.
36
7.
64
8.
9.
196
10
1
11.
4
12.
225
13.
16
14.
169
15.
121
16.
49
100
17.
100
81
18.
25
16
19.
4
25
20.
64
9
21.
49
121
22
121
36
23.
196
81
24.
36
169
27.
12
27
28.
18
50
49
144
See examples 7 8 and 9 on the previous page:
25.
2
98
Chapter 12
26
50
32
© 2015 Eitel
Page 4
Addition and Subtraction Involving Square Roots of Perfect Squares
The operations of Addition and Subtraction can be applied to problems involving square roots. In the
examples below you CANNOT add the square roots together. You must simplify each
square root first and then combine the numbers.
Example 1
Example 2
Example 3
25 + 49
first simplify
each square root
= 5+ 7
= 12
36 + 4
first simplify
each square root
= 6+2
=8
25 − 100
first simplify
each square root
= 5 − 10
= −5
Simplify each expression
1.
100 + 81
2.
16 − 121
3.
1 + 25
4.
64 − 36
5.
9− 4
6.
81 + 16
7.
100 − 49
8.
16 − 49
12–4
1. 19
Chapter 12
2. –7
3. 6
4. 2
5. 1
6. 13
7. 3
8. –3
© 2015 Eitel