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

Examples

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