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``` To simplify a radical in
a perfect square as
a factor
Example: √729
√9 ∙√81
3∙9
Perfect Square! 27
Part I
Vocabulary and Key Concepts
Index
x
Read “the square root of x.”
NOTE: The index 2 is usually omitted when
writing square roots.
Table of Perfect Squares
You may find the following table of perfect
squares to be helpful when you are required
to simplify square roots.
Complete the table below:
1
_____
4
2
2 = _____
9
32 = _____
36
= _____
49
2
7 = _____
64
82 = _____
121
_____
144
2
12 = _____
169
132 = _____
256
_____
289
2
17 = _____
324
182 = _____
12 =
62
112 =
162 =
16
42 = _____
81
92 = _____
196
142 = _____
361
192 = _____
52 = _____
25
102 = _____
100
152 = _____
225
202 = _____
400
ALERT! Check to be sure you have
simplified
completely:
Simplifying
Square Roots
MENTAL MATH: Find two factors of 72, one of which is the
greatest perfect square factor. Establish order, so that
you don’t omit any!
1, 72
2,36
3, 24
4, 18
6, 12
72
36 g2
6 2
8, 9
9,8 (once you have a repeated factor pair, you know that
you have found ALL factors!)
Simplifying Square Roots: An
Alternate Method
72
8g9
NOTE: If you have a
perfect square
(or perfect square factor)
remaining under the
you have not simplified
completely.
4g2g9
2 3 2
6 2
Simplifying Square Roots
KEY: L
K for perfect squares or perfect
square factors.
20
18
27
32
NOTE: If you have a perfect square (or perfect
square factor) remaining under the radical symbol,
you have not simplified completely.
 To multiply, then
simplify square
roots when
possible
Part II
Product of Square Roots
PRODUCT OF SQUARE ROOTS
For all real numbers x ≥ 0, y ≥ 0,
√x ∙√x = √x2 = x
√x ∙√y = √x∙y
NOTE: Squaring a number and finding the
square root are inverse operations.
NOTE:
Squaring a number
and Roots
finding the
square
Multiplying
Square
with
root
are inverse
operations.
Common
√3 ∙√3 =
(√3)2 =
__
3
√4 ∙√4 =
(√4)2 =
____
4
__
√5 ∙√5 =
(√5)2 =
____
__
5
(2√3)2 =
4_ ∙ _3=
__
12
(3√5)2 =
9_ ∙
4∙
_
__
45
(2√5)2 =
_5=
_5=
20
__
Multiplying Square Roots with
√2 ∙√10 =
____ =
√20
√4 ∙√20 =
____
√80 =
(2√3) (5√3) =
____ =
10∙3
√9∙2=
3√2
____ ________
____________
√4∙5 = 2√5
____________
√16∙5 = 4√5
30
____________
(3√2) (5√2) =
____
6∙2 =
____________
12
√3 ∙√6 =
(3√2) (2√6) =
(5√3) (√6) =
√18 =
6____
∙ √12 = ____________
5____
∙ √18 = ____________
 To simplify
an expression
containing a quotient
Quotient of Square Roots
QUOTIENT OF SQUARE ROOTS
For all real numbers x ≥ 0, y > 0
x= x
y y
Simplify each expression:
a.
b.
18
6
24
3
ALERT! When you rationalize, you are changing
Rationalizing Denominators:
the form of the number, but not its value.
1

2
1
2
2
g1 
2
2
2
Fraction
Under √
Denominator
Double Check:
1. Fraction under √ ?
Denominator?
Rationalize
3
4
3
6
1
12
7
9
10
5
Summary
A radical expression is in simplest form
when
 each radicand contains no factor, other
than one, that is a perfect square
 the denominator contains no radicals
and
 each radicand contains no fractions.
Final Checks for Understanding
1. Simplify: √3 ∙√12
2. Simplify: √2 ∙√32
3. Indicate why each expressions is not in