Download Simplifying Radicals

Introduction to Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
radical sign
index
n
a
radicand
This symbol is the radical or the radical sign
 The expression under the radical sign is the
radicand.
 The index defines the root to be taken.
Square Roots
A square root of any positive number has two roots –
one is positive and the other is negative.
If a is a positive number, then
a is the positive (principal) square
root of a and
 a is the negative square root of a.
Examples:
100  10
 0.81   0.9
 36  6
25 5

49 7
1 1
9  non-real #
What does the following symbol represent?
The symbol represents the positive or
principal root of a number.
What is the radicand of the expression 4 5xy ?
5xy
What does the following symbol represent?

The symbol represents the negative root of
a number.
What is the index of the expression
3
3
5x2 y5 ?
What numbers are perfect
squares?
1•1=1
2•2=4
3•3=9
4 • 4 = 16
5 • 5 = 25
6 • 6 = 36
49, 64, 81, 100, 121, 144, ...
Perfect Squares
1
64
225
4
81
256
9
16
100
121
289
25
36
49
144
169
196
400
324
625
4
=2
16
=4
25
=5
100
= 10
144
= 12
Simplifying Radicals
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
48
=
16 * 3 =
4 3
80
=
16 * 5 =
4 5
50
=
25 * 2 =
5 2
125
=
25 * 5 =
5 5
450
=
225 * 2 = 15 2
Simplify
1.
2.
3.
4.
2 18
.
3 8
6 2
36 2
.
.
.
72
Simplify
1. 3x6
2. 3x18
3. 9x6
18
4. 9x
9x
36
+
To combine radicals: combine
the coefficients of like radicals
Simplify each expression
6 7 5 7 3 7 
8 7
5 6 3 7 4 7 2 6 
3 6 7 7
Simplify each expression: Simplify each radical first and
then combine.
2 50  3 32  2 25 * 2  3 16 * 2 
2 * 5 2  3* 4 2 
10 2  12 2 
2 2
Simplify each expression: Simplify each radical first and
then combine.
3 27  5 48  3 9 * 3  5 16 * 3 
3*3 3  5* 4 3 
9 3  20 3 
29 3
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
18
=
=
288
=
=
75
=
=
24
=
=
72
=
=
Simplify each expression
6 5 5 6 3 6 
3 24  7 54 
2 8  7 32 
Simplify each expression
6 5  5 20 
18  7 32 
2 28  7  6 63 
*
To multiply radicals: multiply the
coefficients and then multiply
the radicands and then simplify
the remaining radicals.
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  8 100 
8 *10  80
 5
2

5* 5 
25  5

7* 7 
49  7

8* 8 
64  8

x* x 
x 
 7
2
 8
2
 x
2
2
x
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
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
X
Y
4
2
=X
= Y3
6
6
2
P X Y
4
4X Y
8
2
= P2X3Y
= 2X2Y
10 =
25C D
5C4D5
X
3
=
X
=
Y
5
2
X
*X
X
=
Y
=
2
Y
4
Y
Y
Classwork:
Packet in Yellow Folder under the desk
--- 2nd page
Homework:
worksheet --- Non-Perfect Squares
(#1-12)