6-1 operations on functions - jgray's classroom
TRANSCRIPT
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