Simplifying Radical Expressions
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 = 36⋅2 = 36⋅ 2
=6 2
= 16⋅3 = 16⋅ 3 48
=4 3
Examples:
1. 30a34 = a34 ⋅ 30
= a17 30
2. 54x4 y5z7 = 9x4 y4z6 ⋅ 6yz
=3x2 y2 z3 6yz
Examples:
= 27a3b73 ⋅ 2b3
= 4y2 ⋅ 15xy
=2 y 15xy
3. 54a3b73
4. 60xy3
=3ab2 ⋅ 2b3
Quotient Property of Radicals For any numbers a and b where and , a≥0 b≥0
ab
=a
b
Examples:
1. 716
2. 3225
=
7
16 =
74
=
32
25 =
325
=4 2
5
Examples:
=
483 = 16
=
45
4 =
452
=3 52
3. 48
3
4. 454
=4
Rationalizing the denominator
53
Rationalizing the denominator means to remove any radicals from the denominator.
Ex: Simplify
=
5
3 ⋅
3
3 =
5 3
9 =
153
=5 33
Simplest Radical Form
•No perfect nth power factors other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
Examples:
1. 54
2. 20 8
2 2
=
5
4 =
52
=10
82 =10 4 =10⋅2
=20
Examples:
3.
5
2 2 ⋅
2
2 =
5 22⋅2
=
4 35x
49x2 =
4 5
7x
=
5 24
=5 2
2 4
⋅
7x
7x
=
4 35x7x
4. 4
57x
Adding radicals
6 7+5 7−3 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 6+5 6
=4 6
Multiplying radicals - Distributive Property
3 2+4 3( )
= 3⋅ 2+ 3⋅4 3
= 6+12
Multiplying radicals - FOIL
3+ 5( ) 2+4 3( )
= 6+12+ 10+4 15
= 3⋅ 2+ 3⋅4 3
+ 5⋅ 2+ 5⋅4 3
F O
I L
Examples:
1. 2 3+4 5( ) 3+6 5( )
=6+12 15+4 15+120
=2 3⋅ 3+2 3⋅6 5
+4 5⋅ 3+4 5⋅6 5
F O
I L
=16 15+126
Examples:
2. 5 4 +2 7( ) 5 4−2 7( )
=10⋅10−10⋅2 7
+2 7⋅10+2 7⋅2 7
F O
I L
= 5⋅2+2 7( ) 5⋅2−2 7( )
=100−20 7+20 7−4 49
=100−4⋅7=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+2⋅5
3( )2
−52
=
3+7 3+103−25
=13+7 3
−22
Examples:
⋅6+ 5
6+ 5 2.
1+2 5
6− 5
=6+ 5+12 5+10
62 − 5( )2
=
16+13 531