5.5 Roots of Real Numbers and

Radical Expressions

Definition of nth RootFor any real numbers a and b and any positive integers n,

if an = b,

then a is the nth root of b.

** For a square root the value of n is 2.

Notation

814index

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

Principal Root

The nonnegative root of a number

64

64

64

Principal square root

Opposite of principal square root

Both square roots

Summary of Roots

even

odd

one + root one - root

one + root no - roots

no real roots

no + roots one - root

one real root, 0

Examples

1. 169x4

2. - 8x-3 4

Examples

3. 125x63

4. m3n33

Taking nth roots of variable expressions

Using absolute value signs

If the index (n) of the radical is even, the power under the radical

sign is even, and the resulting power is odd, then we must use

an absolute value sign.

Examples

1. an 44

2. xy2 66

EvenEven

EvenEven

anOdd

=x y2

Odd

3. x6

4. 3 y2 186

EvenEven

x3

Odd

= 3- y2 3

Odd

Even

Even

2

= 3- y2 3

Product Property of Radicals

For any numbers a and b where and , a0

ab a b

b0

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

..\..\Algebra II Honors 2007-2008\Chapter 5\5.5 Simplifying Radicals\Simplifying Radicals.notebook

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 , a0 b0

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 4 102

20

Examples:

3.

5

2 2

2

2

5 222

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

65 3 7

We can only combine terms with radicals if we have like radicals

8 7

Reverse of the Distributive Property

Examples:

1. 23+5+7 3-2

=2 3+7 3+5-2

=9 3+3

Examples:

2. 56 3 24 150

=5 6 3 4 6 25 6

=5 6 6 65 6

=4 6

Multiplying radicals - Distributive Property

3 24 3

3 2 34 3

612

Multiplying radicals - FOIL

3 5 24 3

612 104 15

3 2 34 3

5 2 54 3

F O

I L

Examples:

1. 2 3 4 5 36 5

612 15 4 15120

2 3 3 2 36 5

4 5 3 4 56 5

F O

I L

16 15126

Examples:

2. 5 4 2 7 5 4 2 7

1010 102 7

2 710 2 72 7

F O

I L

= 52 2 7 52 2 7

100 20 7 20 7 4 49 100 4772

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.

32

3 5

35

35

3 3 5 32 3 25

3 2 52

37 3 103 25

137 3

22

Examples:

6 5

6 5 2.

1 2 5

6 5

6 512 510

62 5 2

1613 531