Download Simplifying Radicals

Introduction to Radicals
If b 2 = a, then b is a square root of a.
Meaning
Positive
Square Root

Symbol
Example
Negative
Square Root
9 3
 9  3
The positive and
negative square
roots

 9  3
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 root: one of two equal factors of a given number. The radicand is like the
“area” of a square and the simplified answer is the length of the side of the squares.
•
Principal square root: the positive square root of a number; the principal square
root of 9 is 3.
9 3
•
negative square root: the negative square root of 9 is –3 and is shown like
 9  3
•
radical: the symbol
which is read “the square root of a” is called a radical.
•
radicand: the number or expression inside a radical symbol
radicand.
•
perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc…
are perfect squares.
3
--- 3 is the
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
Simplifying Radical
Expressions
Product Property for Radicals
ab  a  b
36  4  9
36  4  9
6  23
100  4  25
10  2  5
Simplifying Radical Expressions
Product Property for Radicals
50  25  2
5 2
• A radical has been simplified when its radicand
contains no perfect square factors.
• Test to see if it can be divided by 4, then 9, then
25, then 49, etc.
• Sometimes factoring the radicand using the
“tree” is helpful.
x x
14
7
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
Steps to Simplify Radicals:
1. Try to divide the radicand into a perfect
square for numbers
2. If there is an exponent make it even by
using rules of exponents
3. Separate the factors to its own square
root
4. Simplify
Simplify:
12
x
x 
2
6
x
6
Square root of a variable to an
even power = the variable to
one-half the power.
Simplify:
y
y
88
44
Square root of a variable to an
even power = the variable to
one-half the power.
Simplify:
x  x x
12 1
13
x  x
12
x
6
x
Simplify:
50 y
7
25 y  2 y
6
5y
3
2y
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 
Homework
radicals 1
• Complete problems 1-24 EVEN from
worksheet
*
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)
Homework
radicals 2
• Complete problems 1-15 from worksheet.