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