Students will be able to estimate a square root, simplify
a square root, and add and multiply square roots.
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
What is a perfect Square?
A perfect square is the number that
represents the area of the square.
2x2=4
or
2 4
2
The perfect
square is 4

2
2
5 x 5 = 25
OR
5  25
2
5

The perfect
square is
25.
5
The inverse of squaring a number
is to take the square root of the
number.
Think of it as you are given the area
of a square, how long is each side.
The square root of 4 is
2
The square root of 16 is
4
Perfect Squares
(Memorize)
64
225
1
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
625
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
Square
root
1
1 1
81
81  9
4
4 2
100
100 10
9
93
121
121 11
Perfect
square
16
25
36


16  4
144
144 12

25  5
169
169 13
36  6

196
196 14
49  7

225
64  8



49

64

225 15
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
 12.570

158

Obj: Estimate 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
Complete Text Book p 385
A.
t  9.5  90.25
B.
t  8.5  72.25
C
.
t  7.6  57.76
D
.
2
2
2
The tension increases as
the wave speed
increases
Text p 386
1.
v  81
2
2.
The wave speed must be 9 because the
square root of 81 is 9.
v 2  36 The wave speed must be 6 because the
3.
4.
square root of 36 is 6.
Yes -9 because (-9)(-9) is also 81.
Yes -6 because (-6)(-6) is also 36
4 2
 25  -5
 100  -10
49  7
Text p 388
10.
9  13  16
30
25  30  36
25  30  36
9  13  16


11.
13
3  13  4
13  3.6

5  30  6

30  5.5

12.

75
64  75  81
75  8.7
64  75  81

8  75  9

Steps To Simplify Radicals
To SIMPLIFY means to find another expression with the
same value. It does NOT mean to find the decimal
approximation.
8
Step 1: Find the LARGEST PERFECT SQUARE that will divide evenly
into the number under the radical sign. That means when you divide,
you get no remainders, no decimals, no fractions. Perfect square 4
8/4=2

Step 2: Write the number appearing under the radical sign as the
product (multiplication) of the perfect square and your answer from
dividing.
8  4 *2
Step 3: Give each number in the product its own radical sign.
8 4* 2
Step 4:
Reduce the “perfect” radical that you have now
created.
8 2 2
4
=2
16
=4
25
=5
100
= 10
144
= 12
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
8
=
4*2 =
2 2
20
=
4*5
=
2 5
32
=
16 * 2 =
4 2
75
=
25 * 3 =
5 3
40
=
4 *10 = 2 10
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
18
=
9* 2
=
288
=
144 * 2
=
12 2
75
=

25 * 3

=
5 3
24
=
4* 6
=
2 6
72
=
36 * 2
=
6 2






3 2
Simplify
3 50
Don’t let the number in front of the radical
distract you. It is simply “along for the ride” and
will be
multiplied by our final answer
3 50  3 25 * 2
3 25 * 2
3* 5 2
15 2
Perfect Square Factor * Other Factor
LEAVE IN RADICAL FORM
3
5

48
= 3 16 * 3 = 12
3
80
= 5 16 * 5 = 20
5
50

=
25 * 2 = 25 2


7 125 = 7 25 * 5 = 35 5
450
=
225 * 2 = 15 2
Multiplying Radicals
2 6 *5 8
Step 1: Multiply the numbers under the
radical and multiply the numbers
outside the radical.
10 48


Step 2: Simplify if possible
10 16 * 3
10 * 4 3
40 3

Multiply and then simplify
5 * 35  175  25 * 7  5 7
2 8 * 3 7  6 56  6 4 *14 
6 * 2 14  12 14
2 5 * 4 20  20 100  20 *10  200
Simplify the following expressions
-4
764
=
+9
-2
7  8+9
=
=
5 25
+ 49
56 + 9
=
=
= 65
5  5+7
25 + 7
= 32
 5 
5* 5 
25 
 7 
7* 7 
49  7
 8 
8* 8 
64  8
2
2
2
 
x
2
 x* x 
x 
2
5
x
14
2
54
9
7 6
To divide radicals:
divide the
coefficients, divide
the radicands if
possible, and
rationalize the
denominator so that
no radical remains in
the denominator
56

7
8
4* 2  2 2