Transcript perfect squares

Chapter 4
Lesson 4.5
By definition 25 is the number you would
multiply times itself to get 25 for an answer.
Because we are familiar with multiplication,
we know that 25 = 5
Numbers like 25, which have whole numbers
for their square roots, are called perfect
squares
You need to memorize at least the first 15
perfect squares
Square
root
Perfect
square
1
1 = 1
81
4
4 = 2
100
100 = 10
9
9 = 3
121
121 = 11
16
16 = 4
144
144 = 12
25
25 = 5
169
169 = 13
36
36 = 6
196
196 = 14
49
49 = 7
225
225 = 15
64
64 = 8
Perfect
square
Square
root
81
= 9
Every whole number has a square root
Most numbers are not perfect squares, and so
their square roots are not whole numbers.
Most numbers that are not perfect squares
have square roots that are irrational numbers
Irrational numbers can be represented by
decimals that do not terminate and do not repeat
The decimal approximations of whole numbers
can be determined using a calculator
Obj: To find the square root of a number
• Find the square roots of the given numbers
• If the number is not a perfect square, use a
calculator to find the answer correct to the
nearest thousandth.
81
81 = 9
37
37  6.083
158
158  12.570
Obj: To find the square root of a number
• Find two consecutive whole numbers that the
given square root is between
• Try to do this without using the table
18
16 = 4 and 25 = 5
so
18 is between 4 and 5
115
100 = 10 and 121 = 11
so
115 is between 10 and 11
Multiplying radicals
The product of the square roots of two numbers is
the same as the square root of the product of the
numbers
Examples:
3  12
=
36
7  11
=
77
=6
Simplify the following expressions
-4
=
-(2)
764 + 9 = 7  8 + 9
= 56 + 9 = 65
525 + 49 = 5  5 + 7
= 25 + 7 = 32
Simplify the following expressions

4
81

4
=
81
1
–
36

=
1
144
2
9
1
6
1
–
12
2
=
12
1
–
12
=
1
=
12
Rationalizing the Denominator
When the radical in the denominator is not a
perfect square
A form of 1
A form of 1
A form of 1
Simplify
integers
Simplify
integers
Simplified radical form
No factor inside the radical should be a perfect square.
18
= 9  2 = 9 2 = 3 2
108 = 36  3 = 36 3 = 6 3
96
= 16  6 = 16 6 = 4 6
Graphing real numbers
The graph of a number is a dot placed where the
number would be on the number line
Graph the number:
-10
-5
1
32
0
5
10
5
10
Graph the number: -8.5
-10
-5
0
Graphing inequalities
The symbol, [ , means that the point is included in the graph
The symbol, ( , means that the point is not included in the graph
Graph the real numbers between -4 and 6
(
-10
-5
)
0
5
10
Graph the real numbers less than 8
)
-10
-5
0
5
10
Graphing inequalities
The symbol, [ , means that the point is included in the graph
The symbol, ( , means that the point is not included in the graph
Graph the inequality: x > -3
(
-10
-5
0
5
10
5
10
Graph the inequality: x  4
]
-10
-5
0