6-1 Operations on Functions

Operation Definition Example Let ( ) 2f x x and ( ) 5g x x

Addition ( )( ) ( ) ( )f g x f x g x

Subtraction ( )( ) ( ) ( )f g x f x g x

Multiplication ( )( ) ( ) ( )f g x f x g x

Division ( )( )

( )

f f xx

g g x

Examples

Given f(x) = 119961120784 minus 120786 and g(x) = 120784119961 + 120783 find each function Identify any restrictions on the

domain

1) (119891 + 119892)(119909) 2) (119891 ndash 119892)(119909)

3) (119891 119892)(119909) 4) (119891

119892) (119909)

Given f(x) = 120785119961120784 minus 120784119961 + 120783 and g(x) =119961 minus 120786 find each function Identify any restrictions on the

domain

5)(119891 + 119892)(119909) 6) (119891 ndash 119892)(119909)

7) (119891 119892)(119909) 8) (119891

119892) (119909)

Composition of Functions ( )( ) ( ( ))f g x f g x f of g of x

Given f(x) = 120784119961 minus 120787 and g(x) = 120786119961 findhellip

9) ( )( )f g x 10) ( )( )g f x

11) ( )( )f f x 12) ( )( )g g x

Given ( ) 2 3f x x and 2( )g x x find each value

13) ( (3))f g 14) ( (3))g f

15) ( ( 5))f g 16) ( ( 5))g f

Given f = (18) (013) (1511) (149) and g = (815) (51) (1014) (90)

find ( )( )f g x and ( )( )g f x if they exist

( )( )f g x

17)

18)

19)

20)

( )( )g f x

21)

22)

23)

24)

6-2 Inverse Functions amp Relations

Inverse Relation ndash 1) Geometry ndash The vertices of triangle ABC can be represented by the relation (1-1) (6-1) (6-6) Find the inverse of this relation Describe the graph of the inverse Example A ndash (15) (24) (63) Inverse (Arsquo) ndash (51) (42) (36) Find the inverse of each relation

2) 42 1 3 82 3) 30 11 57 99

________________________ ________________________

Inverse Functions

4) Find the inverse of 6

( )2

xf x

Step 1

Step 2

Step 3

Step 4

Graph the function Then write and graph the inverse

5) ( ) 3 1f x x

6 1

( ) 23

f x x

Step 1

Step 2

Step 3

Step 4

Step 1

Step 2

Step 3

Step 4

Determine whether each pair of functions are inverse functions

7) 1

( ) 3 9 ( ) 33

f x x and g x x 8) 2( ) 4 and ( ) 2f x x g x x

9) 1

( ) 3 3 and ( ) 43

f x x g x x 10) 2 1

( ) 2 1 and ( )2

xf x x g x

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Composition of Functions ( )( ) ( ( ))f g x f g x f of g of x

Given f(x) = 120784119961 minus 120787 and g(x) = 120786119961 findhellip

9) ( )( )f g x 10) ( )( )g f x

11) ( )( )f f x 12) ( )( )g g x

Given ( ) 2 3f x x and 2( )g x x find each value

13) ( (3))f g 14) ( (3))g f

15) ( ( 5))f g 16) ( ( 5))g f

Given f = (18) (013) (1511) (149) and g = (815) (51) (1014) (90)

find ( )( )f g x and ( )( )g f x if they exist

( )( )f g x

17)

18)

19)

20)

( )( )g f x

21)

22)

23)

24)

6-2 Inverse Functions amp Relations

Inverse Relation ndash 1) Geometry ndash The vertices of triangle ABC can be represented by the relation (1-1) (6-1) (6-6) Find the inverse of this relation Describe the graph of the inverse Example A ndash (15) (24) (63) Inverse (Arsquo) ndash (51) (42) (36) Find the inverse of each relation

2) 42 1 3 82 3) 30 11 57 99

________________________ ________________________

Inverse Functions

4) Find the inverse of 6

( )2

xf x

Step 1

Step 2

Step 3

Step 4

Graph the function Then write and graph the inverse

5) ( ) 3 1f x x

6 1

( ) 23

f x x

Step 1

Step 2

Step 3

Step 4

Step 1

Step 2

Step 3

Step 4

Determine whether each pair of functions are inverse functions

7) 1

( ) 3 9 ( ) 33

f x x and g x x 8) 2( ) 4 and ( ) 2f x x g x x

9) 1

( ) 3 3 and ( ) 43

f x x g x x 10) 2 1

( ) 2 1 and ( )2

xf x x g x

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-2 Inverse Functions amp Relations

Inverse Relation ndash 1) Geometry ndash The vertices of triangle ABC can be represented by the relation (1-1) (6-1) (6-6) Find the inverse of this relation Describe the graph of the inverse Example A ndash (15) (24) (63) Inverse (Arsquo) ndash (51) (42) (36) Find the inverse of each relation

2) 42 1 3 82 3) 30 11 57 99

________________________ ________________________

Inverse Functions

4) Find the inverse of 6

( )2

xf x

Step 1

Step 2

Step 3

Step 4

Graph the function Then write and graph the inverse

5) ( ) 3 1f x x

6 1

( ) 23

f x x

Step 1

Step 2

Step 3

Step 4

Step 1

Step 2

Step 3

Step 4

Determine whether each pair of functions are inverse functions

7) 1

( ) 3 9 ( ) 33

f x x and g x x 8) 2( ) 4 and ( ) 2f x x g x x

9) 1

( ) 3 3 and ( ) 43

f x x g x x 10) 2 1

( ) 2 1 and ( )2

xf x x g x

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Graph the function Then write and graph the inverse

5) ( ) 3 1f x x

6 1

( ) 23

f x x

Step 1

Step 2

Step 3

Step 4

Step 1

Step 2

Step 3

Step 4

Determine whether each pair of functions are inverse functions

7) 1

( ) 3 9 ( ) 33

f x x and g x x 8) 2( ) 4 and ( ) 2f x x g x x

9) 1

( ) 3 3 and ( ) 43

f x x g x x 10) 2 1

( ) 2 1 and ( )2

xf x x g x

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Determine whether each pair of functions are inverse functions

7) 1

( ) 3 9 ( ) 33

f x x and g x x 8) 2( ) 4 and ( ) 2f x x g x x

9) 1

( ) 3 3 and ( ) 43

f x x g x x 10) 2 1

( ) 2 1 and ( )2

xf x x g x

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-3 Square Root Functions amp Inequalities

Square root function

Radical function

Transformed Square Root Functions ( )f x a x h k

Graph each function State the domain and range

1) ( ) 4f x x 2) ( ) 6 2f x x

x f(x)

Domain ____________Range ______________

x f(x)

Domain ____________Range ______________

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

3) ( ) 2 5f x x 4) ( ) 2 3f x x

Square Root Inequality

5) 4 6y x 6) 2 1y x

7) 2 4y x 8) 4y x

x f(x)

x f(x)

Domain ____________Range ______________

Domain ____________Range ______________

x f(x)

x f(x)

x f(x)

x f(x)

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-4 Nth Roots

Principal root

Numbers and Types of Real Roots

Case Number of Real Roots Example

Odd index

1 real root

Even index positive radicand

2 real roots

Even index negative radicand

0 real roots2 imaginary roots

Radicand of 0

1 root of 0

Simplify

1) 816x 2) 3 63 8x y 3) 2

3 5q 4) 4

5) 1236x 6) 5 10 15243a b 7) 15

3 64 3x 8) 4 816x y

81n

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

9) 2 8( 16)x 10) 16( 7)y

11) 16 20 12100x y z

When you find an even root of an even power and the result is an odd power you must use the

absolute value of the result to ensure that the answer is nonnegative Ex 6 3x x

12) 44 y

13) 18 126 64x y

14) 636y

15) 124 16( 3)x

Designers must create satellites that can resist damage from being struck by small particles of dust and rocks A study showed that the diameter in millimeters d of the hole created in solar cell by dust

particle traveling with energy k in joules is about 30926 0169d k a) Estimate the diameter of the hole created by a particle traveling with energy 35 joules b) If a hole has diameter 25 millimeters estimate the energy with which the particle that made the hole was traveling

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-5 Operations With Radical Expressions

Product Property of Radicals

3 3 32 8 16 or 4 and 3 9 27 or 3

Simplify

1) 832x 2) 24 134 16a b 3) 6 312c d 4) 12 73 27y z

Quotient Property of Radicals

36 6 223

3

27 19 or 3 and or

8 2 23 8

x x xx

Simplify

5) 6

8

x

y 6) 4

4

16

9x

7) 9

5

a

b 8) 3

3

4y

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Multiplying with Radicals

Simplify

9) 3 34 2 25 12 3 18ab a b 10) 3 5 36 8 4 2c d cd

AddingSubtracting Radicals ndash LIKE RADICAL TERMS

Simplify

11) 98 2 32 12) 4 8 3 50 13) 5 12 2 27 128

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Multiply Radicals

Simplify

14) (4 3 5 2)(3 2 6) 15) (6 3 5)(2 5 4 2)

Using a Conjugate to Rationalize a Denominator

CONJUGATE (6 2) (6 2)

16) 2

5 1 17)

3

5 2

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-6 Rational Exponents

Rational Exponent

Rational Exponents

The exponent 1

119899 indicates the 119899119905ℎ root The exponent

119898

119899 indicates the 119899119905ℎ root raised to the 119898119905ℎ power

Example 1

66x x Example 3

334 44x x or x

Write the following in radical form

1) 1

6x 2) 1

5a 3) 7

4d

Write the following in exponential form

4) 4 z 5) 8 c 6) 3 5c

Evaluate each expression

7) 1

481

8) 2

3216 9) 1

53125

10) 3

2( 16)

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

Properties of Rational Exponents

Simplify each expression

11) 2 4

7 7a a 12) 4 27

3 13)

5

6b

14) 3 664z

15)

1

2

1

2

2

3 2

x

x

16)

1

2

1

2

2

2

y

y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

6-7 Solving Radical Equations amp Inequalities

When solving radical equations the result may be a number that does not satisfy the original equation

Such a number is called an ________________________ __________________

Solve each equation

1 4 + radic119909 minus 1 = 5 2 radic3119909 minus 43

= 2 3 6radic119909 + 10 = 42

4 radic2119909 + 14 = 119909 + 3 5 radicminus9119909 + 28 = minus119909 + 4 6 radic119909 minus 12 = 2 minus radic119909

Solving Radical Equations

Steps Example

1 Isolate the radical

radic1199093

minus 2 = 0

2 Raise both sides of the equation to the power equal to the index of the radical

3 Simplify and solve

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12

7 (119909 + 5)13 = 3 8 (2119909 + 15)

12 = 119909 9 3(119909 + 6)

12 = 9

10 radic4119909 + 12 = radic6119909 11 radic7119909 + 12 = 3radic3119909 minus 2 12 radic119909 + 63

= 2radic119909 minus 13

Solve each inequality

13 radic119909 minus 3 + 2 le 5 14 3 + radic5119909 minus 10 le 8 15 radic2119909 + 2 + 1 ge 5

16 radic4119909 minus 4 minus 2 lt 4 17 minus4radic119909 + 3 lt minus12