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9.2 SIMPLIFYING
RADICALS
 Use properties of radicals to
simplify radicals
FINDING SQUARE ROOTS
We can think “what” times “what”
equals the larger number.
36
6
6
= ___
x ___
Is there another answer?
-6
X
-6
SO ±6 IS THE SQUARE ROOT OF 36
FINDING SQUARE ROOTS
We can think “what” times “what” equals
the larger number.
256
Is there another answer?
16
= ___
x 16
___
-16
x
-16
SO ±16 IS THE SQUARE ROOT OF 256
PERFECT SQUARE
 A number that is the square of a whole number
 Can be represented by arranging objects in a
square.
PERFECT SQUARES
PERFECT SQUARES
1x1=1
2x2=4
3x3=9
 4 x 4 = 16
PERFECT SQUARES
1x1=1
2x2=4
3x3=9
 4 x 4 = 16
Activity:
Calculate the perfect
squares up to 152…
PERFECT SQUARES
1x1=1
 9 x 9 = 81
2x2=4
 10 x 10 = 100
3x3=9
 11 x 11 = 121
 4 x 4 = 16
 12 x 12 = 144
 5 x 5 = 25
 13 x 13 = 169
 6 x 6 = 36
 14 x 14 = 196
 7 x 7 = 49
 15 x 15 = 225
 8 x 8 = 64
ESTIMATING
SQUARE ROOTS
25 = ±5
ESTIMATING
SQUARE ROOTS
- 49 = ?
ESTIMATING
SQUARE ROOTS
- 49 = -7
IF THERE IS A SIGN OUT FRONT OF THE RADICAL
THAT IS THE SIGN WE USE!!
ESTIMATING
SQUARE ROOTS
27 = ?
ESTIMATING
SQUARE ROOTS
27 =
27
Since 27 is not a perfect square, we
will leave it in a radical because that
is an EXACT ANSWER.
If you put in your calculator it would
give you 5.196, which is a decimal
apporximation.
ESTIMATING
SQUARE ROOTS
Not all numbers are perfect
squares.
Not every number has an Integer
(whole number) for a square root.
We have to estimate square roots
for numbers between perfect
squares.
Simplifying Radicals
Properties of Radicals
Product Property
Example:
Quotient
Property
ab  a  b
4 100  4  100
a
a

b
b
Example:
9
9

25
25
An expression with radicals is in simplest form
if the following are true:
• No perfect square
factors other than 1
are in the radicand
8  4 2  4  2  2 2
• No fractions are in
the radicand
5
5
5


4
16
16
• No radicals appear in
the denominator of a
fraction
1
1

4 2
Example 1 Simplify the expression
Factor using perfect square factors.
Use the product property.
Simplify.
50
50  25  2
 25  2
5 2
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
Example 2 Simplify the expression.
a.
3
4
b.
18
25
c.
40
90
3
4
18
25
10  4
10  9
3
2
18
5
4
9
9 2
5
4
9
3 2
5
2
3
Divide out common
factors. 10’s cancel
Example
Simplify
60
4  15
2 15
Simplify
40
2 10
4  10


36
6
6
1 10

3
10
3
This cannot be divided
which leaves the radical
in the denominator. We
do not leave radicals in
the denominator. So we
need to rationalize by
multiplying the fraction
by something so we
can eliminate the
radical in the
denominator.
6

7
6
*
7
42

49
7

7
42
7
42 cannot be simplified,
so we are finished.
This can be divided
which leaves the radical
in the denominator. We
do not leave radicals in
the denominator. So we
need to rationalize by
multiplying the fraction
by something so we
can eliminate the
radical in the
denominator.
5

10
1
*
2
2
2
2

2
This cannot be divided
which leaves the radical
in the denominator. We
do not leave radicals in
the denominator. So we
need to rationalize by
multiplying the fraction
by something so we
can eliminate the
radical in the
denominator.
3

12
3
*
12
3

3
3 3

36
Reduce
the
fraction.
3 3

6
3
2
ASSIGNMENT
9.2 (pg. 514)
#10-48 EVEN