5.6 Radical
Expressions
Rationalizing the
denominator
Like radical expressions
Conjugates
Product Property of Radicals
If you have the same index, then you can
multiply the radicands.
7  5  35
3
4  3 5  3 20
Product Property of Radicals
Also, go the other way, the simplify radical
expressions
98  2  49
7 2
Simplify
4 9
25a b
Simplify
4 9
25a b
25  a  b  b
4
2 4
5a b
b
8
Quotient Property of Radicals
Quotients can be broken apart.
1
1 1


25
25 5
Some rules about Radicals
Reduce them to the lowest index.
4
4 2  2
4
2
Some rules about Radicals
Radicals can not have fractions in them.
This is called Rationalizing the denominator.
2
2 5


5
5 5
10
10
10



25
5
25
Some rules about Radicals
Radicals can not be in the denominator.
No tents in the basement.
4
4
3 4 3 4 3




3
3
3 3
9
Simplify
y8
x7
Simplify
y8
7
x
y8 x
 
7
x x
xy8
8
x
Simplify
8
y
7
x
y8 x
 
7
x x

xy8
x8
xy8
8
x
y4 x

x4
Simplify
3
2 3 2 3x 2
 2  2
9x
3 x 3x
Simplify
3
2 3 2 3x 2
 2  2
9x
3 x 3x
2
2
3
6
x
6
x
3 3 3 
3 3 3
3 x
3 x
Simplify
3
2 3 2 3x 2
 2  2
9x
3 x 3x
3
2
2
6
x
6
x
3 3 3 
3 3 3
3 x
3 x
3
6x2

3x
Multiply Radicals
You can only multiply radicals if they have
the same index.
5 100a  10a  5 1000a
3
2
3
3
 53 103 a 3  5  10  a  50a
3
Adding and Subtract Radicals
If radical are to the same index and have the
same radicand, then they are like radical
expressions. Like radicals can be added or
subtracted.
53 7  63 7  113 7
56 12  76 12  26 12
Simplify
Simplify radicals first.
3 45  5 80  4 20
Simplify
Simplify radicals first.
3 45  5 80  4 20
3 9  5  5 16  5  4 4  5
Simplify
Simplify radicals first. 3 45  5 80  4 20
3 9  5  5 16  5  4 4  5
33 5  5 4 5  4  2 5
Simplify
Simplify radicals first. 3 45  5 80  4 20
3 9  5  5 16  5  4 4  5
33 5  5 4 5  4  2 5
9 5  20 5  8 5  3 5
Multiply Radicals
Remember foiling
2

3 3 5 3 3

Multiply Radicals
Remember foiling
2

3 3 5 3 3




3 2 3  2 3   3  33 5  3 5  3

Multiply Radicals
Remember foiling
2

3 3 5 3 3




3 2 3  2 3   3  33 5  3 5  3
6 3  2 9  9 5  3 15

Multiply Radicals
Remember foiling
2

3 3 5 3 3




3 2 3  2 3   3  33 5  3 5  3

6 3  2 9  9 5  3 15
6 3  2(3)  9 5  3 15  6 3  6  9 5  3 15
Simplify
4

2 7 4 2 7

Simplify
4

2 7 4 2 7

16 4  28 2  28 2  49
16(2)  49  32  49  17
Conjugate
You can not have radicals in the denominator. We
have to use the conjugate to rational the
denominator in fractions.
3
4 5
Conjugate
You can not have radicals in the denominator. We
have to use the conjugate to rational the
denominator in fractions.
3
4 5


3
4 5

4 5 4 5


Conjugate
Conjugate
You can not have radicals in the denominator. We
have to use the conjugate to rational the
denominator in fractions.
3
4 5


3
4 5

4 5 4 5


12  3 5 12  3 5

11
16  25
Simplify
Use the conjugate
2 3
4 3
Simplify
Use the conjugate
2 3
4 3


2 3 4 3

4 3 4 3


8  2 3  4 3  (3)
16  4 3  4 3  3
11  6 3 11  6 3

16  3
13
Homework
Page 254 – 255
#15 – 49 odd
57,58
Homework
Page 254 – 256
# 16 – 48 even
75 - 82