Objectives
The student will be able to:
1. simplify square roots, and
2. simplify radical expressions.
If x2 = y then x is a square root of y.
In the expression 64 ,
is the radical sign and
64 is the radicand.
1. Find the square root: 64
8
2. Find the square root:  0.04
-0.2
3. Find the square root:  121
11, -11
4. Find the square root:
21
5. Find the square root:
5

9
441
25

81
6. Use a calculator to find each
square root. Round the decimal
answer to the nearest hundredth.
 46.5
6.82, -6.82
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, ...
1. Simplify
147
Find a perfect square that goes into 147.
147  49 3
147  49
147  7 3
3
2. Simplify 605
Find a perfect square that goes into 605.
121 5
121
11 5
5
Simplify
1.
2.
3.
4.
2 18
.
3 8
6 2
36 2
.
.
.
72
How do you simplify variables in the radical?
x
7
Look at these examples and try to find the pattern…
x  x
2
x x
3
x x x
4
2
x x
5
2
x x x
6
3
x x
1
What is the answer to
x x
7
3
x ?
7
x
As a general rule, divide the
exponent by two. The
remainder stays in the
radical.
4. Simplify 49x
2
Find a perfect square that goes into 49.
49 x
7x
2
5. Simplify 8x
25
4 2x
12
2x 2x
25
Simplify
1.
2.
3.
4.
3x6
3x18
6
9x
18
9x
9x
36
6. Simplify 6  10
Multiply the radicals.
60
4 15
4
15
2 15
7. Simplify 2 14  3 21
Multiply the coefficients and radicals.
6 294
6 49 6
6
49
67
6
6
42 6
Simplify 6 x
1.
2.
3.
4.
4x
.
2
3
4
4 3x
2
x 48
4
48x
.
.
.
3
8x
How do you know when a radical
problem is done?
1. No radicals can be simplified.
Example:
8
2. There are no fractions in the radical.
1
Example:
4
3. There are no radicals in the denominator.
Example:
1
5
8. Simplify.
Whew! It
simplified!
108
3
Divide the radicals.
108
3
36
6
Uh oh…
There is a
radical in the
denominator!
8
2
9. Simplify
2 8
4 1
4
Whew! It simplified
again! I hope they
all are like this!
4
2
2
Uh oh…
Another
radical in the
denominator!
10. Simplify
5
7
Uh oh…
There is a
fraction in
the radical!
Since the fraction doesn’t reduce, split the radical up.
5
7
5

7
How do I get rid
of the radical in
the denominator?
7
7
35

49
Multiply by the “fancy one”
to make the denominator a
perfect square!
35

7
Product Property of
Radicals
For any numbers a and
b where a 0and b 0,
ab  a  b
Product Property of
Radicals Examples
72 
36 2  36 2
6 2
48  16 3  16 3
4 3
Examples:
1.
30a  a  30
34
34
 a
17
2.
30
54x y z  9x y z  6yz
4 4 6
4 5 7
 3x y z
2
2
3
6 yz
Examples:
3.
3
54a b  27a b  2b
3 7
3
3 7
3
 3ab  2b
2
4.
3
60xy  4 y  15xy
3
2
 2 y 15xy
Quotient Property of
Radicals
For any numbers a and
b where a 0and b 0,
a

b
a
b
Examples:
1.
7

16
32

2.
25
7
7

4
16
32
25

32 4 2

5
5
Examples:
3.
4.
48
3

45

4
48

3
16  4
45
45 3 5

2
2
4

Simplest Radical Form
•No perfect nth power factors
other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
Adding radicals
We can only combine terms with radicals
if we have like radicals
6 7 5 7  3 7
 6 5 3 7  8 7
Reverse of the Distributive Property
Examples:
1. 2 3 + 5 + 7 3 - 2
= 2 3 + 7 3 + 5- 2
= 9 3+3
Examples:
2. 5 6  3 24  150
= 5 6  3 4 6  25 6
= 5 6 6 6  5 6
=4 6
Multiplying radicals Distributive Property
 2  4 3
3

3 2  3 4 3

6  12
Multiplying radicals - FOIL

3 5
F


24 3
O
3 2  3 4 3
I

L
 5 2  5 4 3

6  12 10  4 15

Examples:


1. 2 3  4 5 3  6 5
O
F
 2 3 3  2 3 6 5
L
I
4 5  3  4 5  6 5
 6  12 15  4 15  120
 16 15  126

Examples:
5 4  2 7
= 5 2 2 75 2 2 7
O
F
2. 5 4  2 7
 1010 10 2 7
I
L
2 7 10 2 7  2 7
 100 20 7  20 7  4 49
 100 4 7  72
Conjugates
Binomials of the form
a b  c d anda b  c d
where a, b, c, d are rational
numbers.
Ex:
5  6  Conjugate:
56
3  2 2  Conjugate: 3  2 2
What is conjugate of
2 7  3?
Answer: 2 7  3
The product of conjugates is a
rational number. Therefore, we can
rationalize denominator of a fraction
by multiplying by its conjugate.
Examples:
32 35

1.
35 35
3  3  5 3  2 3  2 5

2
2
3 5
3  7 3  10 13 7 3


22
3 25
 
Examples:
1 2 5 6  5

2.
6 5 6  5

6  5  12 5  10
6 
2
 
2
5
16 13 5

31