5.5 Roots of Real
Numbers and
Radical
Expressions
Notation
814
index radical
radicand
Note: An index of 2 is understood but not
written in a square root sign.
Simplify 814
To simplify means to find x
in the equation:
x4 = 81
Solution: = 3 814
Examples
1. 169x4
2. - 8x- 3 4
Examples
3. 125x63
4. m3n
33
Product Property of
Radicals
For any numbers a and
b where and ,
a 0
ab a b
b 0
Product Property of
Radicals Examples
72 362 36 2
6 2
163 16 3 48
4 3
What to do when the index will not
divide evenly into the radical????
• Smartboard Examples
• 5.5 Simplifying Radicals with
cards.notebook
Examples:
1. 30a34
a34 30
a
1730
2. 54x4y
5z
7
9x4y
4z
6 6yz
3x
2y
2z
36yz
Examples:
27a3b
73 2b
3
4 y2 15xy
2 y 15xy
3. 54a3b
73
4. 60xy3
3ab2 2b
3
Quotient Property of
Radicals
For any numbers a and
b where and ,
a 0 b 0
a
b
a
b
Examples:
1. 7
16
2. 32
25
7
16
7
4
32
25
32
5
4 2
5
Examples:
48
3 16
45
4
45
2
3 5
2
3. 48
3
4. 45
4
4
Rationalizing the
denominator
5
3
Rationalizing the denominator means
to remove any radicals from the
denominator.
Ex: Simplify
5
3
3
3
5 3
9
15
3
5 3
3
Simplest Radical Form
•No perfect nth power factors
other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
Examples:
1. 5
4
2. 20 8
2 2
5
4
5
2
10
8
2 10 4 102
20
Examples:
3.
5
2 2
2
2
5 2
22
4 35x
49x2
4 5
7x
5 2
4
5 2
2 4
7x
7x
4 35x
7x
4. 4
5
7x
Adding radicals
6 7 5 73 7
6 5 3 7
We can only combine terms with radicals
if we have like radicals
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 65 6
= 4 6
Multiplying radicals -
Distributive Property
3 2 4 3
3 2 3 4 3
612
Multiplying radicals - FOIL
3 5 2 4 3
612 104 15
3 2 3 4 3
5 2 5 4 3
F O
I L
Examples:
1. 2 3 4 5 3 6 5
612 15 4 15120
2 3 3 2 3 6 5
4 5 3 4 56 5
F O
I L
16 15126
Examples:
2. 5 4 2 7 5 4 2 7
1010102 7
2 7 10 2 72 7
F O
I L
= 5 2 2 7 52 2 7
10020 7 20 7 4 49
10047 72
Conjugates
Binomials of the form
where a, b, c, d are rational numbers.
a b c d and a b c d
The product of conjugates is a
rational number. Therefore, we can
rationalize denominator of a fraction
by multiplying by its conjugate.
Ex: 5 6 Conjugate: 5 6
3 2 2 Conjugate: 3 2 2
What is conjugate of 2 7 3?
Answer: 2 7 3
Examples:
1.
3 2
3 5
3 5
3 5
3 3 5 3 2 3 25
3 2
52
3 7 3 10
3 25
13 7 3
22
Examples:
6 5
6 5 2.
1 2 5
6 5
6 5 12 5 10
62 5 2
1613 5
31