Post on 25-Jun-2018
MTH 209 Week 3
Due for this week…
Homework 3 (on MyMathLab – via the Materials
Link) The fifth night after class at 11:59pm.
Read Chapter 6.6, 8.4 and 11.1-11.5
Do the MyMathLab Self-Check for week 3.
Learning team hardest problem assignment.
Complete the Week 3 study plan after submitting
week 3 homework.
Participate in the Chat Discussions in the OLS
Slide 2 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Section 7.1
Introduction to
Rational
Expressions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Basic Concepts
• Simplifying Rational Expressions
• Applications
Basic Concepts
Rational expressions can be written as quotients
(fractions) of two polynomials.
Examples include:
2
3
2
, , 5
4
6
3
4 1
4 8
x x
x
x
xx
Example
If possible, evaluate each expression for the given
value of the variable.
a. b. c.
Solution
a. b. c.
1; 3
3x
x
2
; 43 4
ww
w
4; 5
4
ww
w
1; 3
3x
x
2
;3 4
4w
ww
4;
45
ww
w
1
3
1
3 6
2( )
3
4
( 44)
16
12 4
162
8
4 ( )
5)
5
( 4
91
9
Try Q 7,11,13,17 pg. 427
Example
Find all values of the variable for which each
expression is undefined.
a. b. c.
Solution
a. b. c.
2
1
x
2
4
w
w2
6
4w
2
1
x
2
4
w
w2
6
4w
Undefined
when x2 = 0 or
when x = 0.
Undefined
when w – 4 =
0 or when w =
4.
Undefined
when w2 – 4 =
0 or when w =
2.
Try Q 25,27,31,33 pg. 428
Example
Simplify each fraction by applying the basic principle
of fractions.
a. b. c.
Solution
a. The GCF of 9 and 15 is 3.
b. The GCF of 20 and 28 is 4.
c. The GCF of 45 and 135 is 45.
9
15
20
28
45
135
9
15 5
3
3
3
3
5
20
28 7
4
4
5
5
7
45
135
145
5 34
1
3
Try Q 39,43 pg. 428
Example
Simplify each expression.
a. b. c.
Solution
a. b. c.
2
16
4
y
y
3 12
4 16
x
x
2
2
25
2 7 15
x
x x
2
16
4
y
y
3 12
4 16
x
x
44
4
y
y y
4
y
3( )
4(
4
)4
x
x
3
4
2
2
25
2 7 15
x
x x
( )( 5)
(2
5
3)( )5
x
x
x
x
5
2 3
x
x
Example
Simplify each expression.
a. b.
Solution
a. b.
7
2 14
y
y
8
8
x
x
7
2 14
y
y
1( )
2(
7
7)
y
y
1
2
8
8
x
x
(8 )
8
x
x
8
8
x
x
81
8
x
x
Try Q 51,55,61,79 pg. 428
Example
Suppose that n balls, numbered 1 to n, are placed in
a container and two balls have the winning number.
a. What is the probability of drawing the winning ball
at random?
b. Calculate this probability for n = 100, 1000 and
10,000.
c. What happens to the probability of drawing the
winning ball as the number of balls increases?
Solution
a. There are 2 chances of drawing the winning ball.
2
n
Try Q 105 pg. 429
Example (cont)
b. Calculate this probability for n = 100, 1000 and
10,000.
c. What happens to the probability of drawing the
winning ball as the number of balls increases?
The probability decreases.
2 1
100 50
2 1
1000 500
2 1
10,000 5000
Try Q 105 pg. 429
Section 7.2
Multiplication
and Division of
Rational
Expressions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Review of Multiplication and Division of Fractions
• Multiplication of Rational Expressions
• Division of Rational Expressions
Example
Multiply and simplify your answers to lowest terms.
a. b. c.
Solution
a.
b.
c.
4 5
9 7
415
5
2 5
7 8
4 5 20
9 7 63
4 15 4 6015 12
5 1 5 5
2 5 5 2 5 1 5
7 8 7 8 7 4 28
Try Q 5,7,9 pg. 435
Example
Divide and simplify your answers to lowest terms.
a. b. c.
Solution
a. b.
c.
1 3
6 5
618
7
4 11
5 15
1 3
6 5
618
7
4 11
5 15
1 5
6 3
5
18
1 6
7 18
6 1
7 18
1
21
60
55
4 15
5 11
12
11
12 5
11 5
Try Q 13,15,17 pg. 435
Example
Multiply and simplify to lowest terms. Leave your
answers in factored form.
a. b.
Solution
a. b.
2
6 5
10 12
x
x
3 4
2 1 3 9
x x
x x
2
6 5
10 12
x
x
210 12
6 5x
x
21
0
0
3
2
x
x
1
4x
3 4
2 1 3 9
x x
x x
( 3)( 4)
(2 1) 3 )9(x x
x x
( )( 4)
3(2 1)( 3)
3 x
x
x
x
4
3(2 1)
x
x
Example
Multiply and simplify to lowest terms. Leave your
answer in factored form.
Solution
2
2
16 3
9 4
x x
x x
2
2
( 3)
(
( )
4)
6
( 9)
1
x
x
x
x
2
2
16 3
9 4
x x
x x
( 3)( 4)( 4)
( 3)( 3 4))(x
xx x
x x
( 4)( )( )
( 3)
3
( 3 )
4
4)(
x
x
x x
x x
( 4)
( 3)
x
x
Try Q 29,31,41,45 pg. 435
Example
Divide and simplify to lowest terms.
a. b.
Solution
a. b.
3 2 1
6
x
x x
2
2
16 4
2 8 2
x x
x x x
3 2 1
6
x
x x
3 6
2 1
x
x x
18
(2 1)
x
x x
18
2 1x
2
2
16
2 8
4
2
x
x x
x
x
2
2
16
42 8
2x
x x
x
x
( 4)( 4) 2
( 2)( 4) 4
x x x
x x x
( )( )( )
( )( )(
4
2 4
24
)4
x
xx
x
x
x
1
Try Q 49,57,65 pg. 435
Section 7.3
Addition and
Subtraction
with Like
Denominators
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Review of Addition and Subtraction of Fractions
• Rational Expressions Having Like Denominators
Example
Simplify each expression to lowest terms.
a. b.
Solution
a.
b.
4 1
7 7
1 5
9 9
4 1
7 7
4 1
7
5
7
1 5
9 9
1 5
9
6
9
2
3
Example
Simplify each expression to lowest terms.
a. b.
Solution
a.
b.
13 7
18 18
15 11
30 30
13 7
18 18
13 7
18
6
18
1
3
15 11
30 30
15 11
30
2
15
4
30
Try Q 7,9,11,13 pg. 442
To add two rational expressions having like
denominators, add their numerators. Keep
the same denominator.
C is nonzero
SUMS OF RATIONAL EXPRESSIONS
A B A B
C C C
Example
Add and simplify to lowest terms.
a. b.
Solution
a.
b.
4 1 2
3 3
x x
x x
2 2
5
7 10 7 10
x
x x x x
3
4
3
1 2
x x
x x
4 1 2
3
x x
x
3
5 1x
x
2 2
5
7 10 7 10
x
x x x x
2
5
7 10
x
x x
5
5 2
x
x x
1
2x
Try Q 19,25,33,35 pg. 442
Example
Add and simplify to lowest terms.
a. b.
Solution
a.
b.
7 4
ab ab
2 2 2 2
w y
w y w y
7 4
ab ab
7 4
ab
11
ab
2 2 2 2
w y
w y w y
2 2
w y
w y
( )( )
w y
w y w y
1
w y
Try Q 51,53,55 pg. 443
To subtract two rational expressions having
like denominators, subtract their numerators.
Keep the same denominator.
C is nonzero
DIFFERENCES OF RATIONAL EXPRESSIONS
A B A B
C C C
Example
Subtract and simplify to lowest terms.
a. b.
Solution
a.
b.
2 2
6 6x
x x
2 2
2 3 4
1 1
x x
x x
2 2
6 6x
x x
2
6 6x
x
2
6 6x
x
2
x
x
1
x
2 2
2 3 4
1 1
x x
x x
2
2 3 4
1
x x
x
1
1 1
x
x x
2
1
1
x
x
1
1x
Try Q 21,27,67 pg. 442-3
Example
Subtract and simplify to lowest terms.
Solution
7 2
2 2
a a
a a
7 2
2 2
a a
a a
7 ( 2)
2
a a
a
6 2
2
a
a
Try Q 31,59 pg. 442-3
Section 7.4
Addition and
Subtraction
with Unlike
Denominators
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Finding Least Common Multiples
• Review of Fractions Having Unlike Denominators
• Rational Expressions Having Unlike Denominators
The least common multiple (LCM) of two or
more polynomials can be found as follows.
Step 1: Factor each polynomial completely.
Step 2: List each factor the greatest number
of times that it occurs in either
factorization.
Step 3: Find the product of this list of factors.
The result is the LCM.
FINDING THE LEAST COMMON MULTIPLE
Example
Find the least common multiple of each pair of
expressions.
a. 6x, 9x4 b. x2 + 7x + 12, x2 + 8x + 16
Solution
Step 1: Factor each polynomial completely.
6x = 3 ∙ 2 ∙ x 9x4 = 3 ∙ 3 ∙ x ∙ x ∙ x ∙ x
Step 2: List each factor the greatest number of
times.
3 ∙ 3 ∙ 2 ∙ x ∙ x ∙ x ∙ x
Step 3: The LCM is 18x4.
Example (cont)
b. x2 + 7x + 12, x2 + 8x + 16
Step 1: Factor each polynomial completely.
x2 + 7x + 12 = (x + 3)(x + 4)
x2 + 8x + 16 = (x+ 4)(x + 4)
Step 2: List each factor the greatest number of times.
(x + 3), (x + 4), and (x + 4)
Step 3: The LCM is (x + 3)(x + 4)2.
Try Q 15,19,27,29 pg. 451
Example
Simplify each expression.
a. b.
Solution
a. The LCD is the LCM, 42.
b. The LCD is 60.
4 1
7 6
5 11
12 30
4 1
7 6
4 6 1 7
7 6 6 7
24 7
42 42
31
42
5 11
12 30
5 5 11 2
12 5 30 2
25 22
60 60
3
60
1
20
Try Q 45,47 pg. 452
Example
Find each sum and leave your answer in factored
form.
a. b.
Solution
a. The LCD is x2.
b.
2
2 5
x x
4 3
1 1x x
2
2 5
x x
2
2 5x
x x x
2 2
2 5x
x x
2
2 5x
x
4 3
1 1x x
4 3 1
1 1 1x x
4 3
1 1x x
1
1x
Try Q 53,65,71 pg. 452
Example
Simply the expression. Write your answer in lowest
terms and leave it in factored form.
Solution
The LCD is x(x + 7).
3 5
7
x
x x
3 5
7
x
x x
3 7 5
7 7
x x x
x x x x
3 7 5
7 7
x x x
x x x x
3 7 5
7
x x x
x x
2 4 21 5
7
x x x
x x
2 21
7
x x
x x
Example
Simplify the expression. Write your answer in lowest
terms and leave it in factored form.
Solution
2 2
6 5
6 9 9x x x
6 5
3 3 3 3x x x x
2 2
6 5
6 9 9x x x
6 5
3 3 3 33
33
3
x
x x xx x
x
x
6 3 5 3
3 3 3 3 3 3
x x
x x x x x x
6 18 5 15
3 3 3
x x
x x x
33
3 3 3
x
x x x
Try Q 63,77,79,81 pg. 452
Example
Add and then find the reciprocal of the result.
Solution
The LCD is RS.
1 1,
R S
1 1 1 1
R S R
R
S
S
S R
S R
RS RS
S R
RS
The reciprocal is
.RS
S R
Try Q 101 pg. 453
Section 7.6
Rational
Equations and
Formulas
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Solving Rational Equations
• Rational Expressions and Equations
• Graphical and Numerical Solutions
• Solving a Formula for a Variable
• Applications
Rational Equations
If an equation contains one or more rational
expressions, it is called a rational equation.
Example
Solve each equation.
a. b.
Solution
a. b.
9 4
7 x
6
2 1x
x
9 4
7 x
9 28x
28
9x
6
2 1x
x
6
2 1 1
x
x
2 1 6(1)x x
22 6x x
22 6 0x x
2 3 2 0x x
2 3 0x
3
2x 2x
The solutions are 3
2 and .2
The solution is
28.
9
2 0x
Try Q 9,15,31,41 pg. 473
Example
Determine whether you are given an expression or
an equation. If it is an expression, simplify it and then
evaluate it for x = 4. If it is an equation, solve it.
a. b.
Solution
a. There is an equal sign, so it is an equation.
24
3 3
x x
x x
2 3 10
2 2
x x
x x
2
3 4 3 33 3
x xx x x
x x
2 4 12x x x
4x
3 12x The answer checks.
The solution is −4.
Example (cont)
b. There is no equals sign, so it is an expression.
The common denominator is x – 2, so we can add
the numerators.
2 3 10
2
x x
x
2 3 10
2 2
x x
x x
2 5
2
x x
x
5x
When x = 4, the expression evaluates 4 + 5 = 9.
2 3 10
2 2
x x
x x
Try Q 49,55 pg. 473
Example
Solve graphically and numerically.
Solution
Graph and
2
1x
x
1
2
1y
x
2 .y x
The solutions are −2 and 1.
x −3 −2 −1 0 1 2 3
−1 −2 −− 2 1 2
3
1
2(−2, −2)
(1, 1)
Example (cont)
Solve graphically and numerically.
Solution
Numerical Solution
2
1x
x
The solutions are −2 and 1.
x −3 −2 −1 0 1 2 3
−1 −2 −− 2 1
−3 −2 −1 0 1 2 3
1
2
1y
x
2
3
1
2
2y x
Try Q 75 pg. 474
Example
Solve the equation for the specified variable.
Solution
2 for C r r
2 for C r r
2
2 2
rC
2
Cr
Example
Solve the equation for the specified variable.
Solution
2 for
Ah A
B b
2h
b
A
B
( ) 2h B b A
( )
2
h B bA
Example
Solve the equation for the specified variable.
Solution
2 for S B sl s
2 for S B ls s
2 S B B B ls
2 S B ls
2
2 2
S B
l
sl
l
2
S Bs
l
Try Q 89,91,97 pg. 474
Example
A pump can fill a swimming pool ¾ full in 6 hours,
another can fill the pool ¾ full in 9 hours. How long
would it take the pumps to fill the pool ¾ full, working
together?
Solution
3
6 9 4
t t
336 36
6 9 4
t t
6 4 27t t
72
10t
The two pumps can fill the pool ¾ full in
hours.
72
10
Try Q 102 pg. 474
Section 7.7
Proportions
and Variation
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Proportions
• Direct Variation
• Inverse Variation
• Analyzing Data
• Joint Variation
Proportions
A proportion is a statement (equation) that two
ratios (fractions) are equal.
The following property is a convenient way to solve
proportions:
a
b d
c is equivalent to ,ad bc
provided b ≠ 0 and d ≠ 0.
Example
On an elliptical machine, Francis can burn 370
calories in 25 minutes. If he increases his work time
to 30 minutes, how many calories will he burn?
Solution
Let x be the equivalent amount of calories.
25 30
370 x
Minutes Minutes=
Calories Calories
25 11,100x
444x Thus, in 30 minutes,
Francis will burn 444
calories.
Try Q 65 pg. 488
Example
A 6-foot tall person casts a shadow that is 8-foot
long. If a nearby tree casts a 32-foot long shadow,
estimate the height of the tree.
Solution
The triangles are similar because
the measures of its corresponding
angles are equal. Therefore corresponding sides are
proportional.
6 ft h
8 ft 32 ft
32
6 8
h
Height Shadow length=
Height Shadow length
8 192h
24h The tree is 24 feet tall.
Try Q 56 pg. 488
Example
Let y be directly proportional to x, or vary directly with
x. Suppose y = 9 when x = 6. Find y when x = 13.
Solution
Step 1 The general equation is y = kx.
Step 2 Substitute 9 for y and 6 for x in
y = kx. Solve for k.
Step 3 Replace k with 9/6 in the equation y = 9x/6.
Step 4 To find y, let x = 13.
9 6
9
6
y kx
k
k
9
6
9(13)
6
19.5
y x
y
y
Try Q 33 pg. 487
Example
The table lists the amount of pay for various hours
worked.
a. Find the constant of proportionality.
b. Predict the pay for 19 hours of work.
Hours Pay
6 $138
11 $253
15 $345
23 $529
31 $713
6, 138
11, 253
15, 345
23, 529
31, 713
0
100
200
300
400
500
600
700
800
0 10 20 30 40
Pa
y (
do
lla
rs)
Hours
Example (cont)
The slope of the line equals the proportionality, k. If
we use the first and last data points (6, 138) and (31,
713), the slope is
The amount of pay per hour is $23. The graph of the
line y = 23x, models the given graph.
To find the pay for 19 hours, substitute 19 for x.
713 138
31 6k
23
y = 23x,
y = 23(19)
y = 437
19 hours of work would
pay $437.00
Try Q 73 pg. 488
Example
Let y be inversely proportional to x, or vary inversely
with x. Suppose y = 6 when x = 4. Find y when x = 8.
Solution
Step 1 The general equation is y = k/x.
Step 2 Substitute 6 for y and 4 for x in
Solve for k.
Step 3 Replace k with 24 in the equation y = k/x.
Step 4 To find y, let x = 8.
64
24
ky
x
k
k
24
8
3
ky
x
y
y
Try Q 39 pg. 487
Example
Determine whether the data in each table represent
direct variation, inverse variation, or neither. For
direct and inverse variation, find the equation.
a.
b.
c.
x 3 7 9 12
y 12 28 32 48
x 5 10 12 15
y 12 6 5 4
x 8 11 14 21
y 48 66 84 126
Neither the product xy nor the ratio y/x
are constant in the data in the table.
Therefore there is neither direct
variation nor indirect variation in this
table. As x increases, y decreases. Because
xy = 60 for each data point, the
equation
y = 60/x models the data. This
represents an inverse variation.
The equation y = 6x models the
data. The data represents direct
variation.
Try Q 51a,53a,55a pg. 487-488
, 0 and 1,xf x a a a
Let x, y, and z denote three quantities.
Then z varies jointly with x and y if
there is a nonzero number k such that
JOINT VARIATION
.z kxy
Example
The strength S of a rectangular beam varies jointly
as its width w and the square of its thickness t. If a
beam 5 inches wide and 2 inches thick supports 280
pounds, how much can a similar beam 4 inches wide
and 3 inches thick support?
Solution
The strength of the beam is
modeled by S = kwt2.
2280 5 2k
280
5 4k
14k
Try Q 83 pg. 488
Example (cont)
Thus S = 14wt2 models the strength of this type of
beam. When w = 4 and t = 3, the beam can support
S = 14 ∙ 4 ∙ 32 = 504 pounds
Try Q 83 pg. 488
Section 10.1
Radical
Expressions
and Functions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Radical Notation
• The Square Root Function
• The Cube Root Function
Radical Notation
Every positive number a has two square roots, one
positive and one negative. Recall that the positive
square root is called the principal square root.
The symbol is called the radical sign.
The expression under the radical sign is called the
radicand, and an expression containing a radical
sign is called a radical expression.
Examples of radical expressions: 5
7, 6 2, and 3 4
xx
x
Example
Evaluate each square root.
a.
b.
c.
36
0.64
4
5
0.8
16
25
6
Try Q 15,17,19,21 pg. 641
Example
Approximate to the nearest thousandth.
Solution
38
6.164
Try Q 39 pg. 641
Example
Evaluate the cube root.
a.
b.
c.
3 64
3 125
1
2
5
31
8
4
Try Q 23,25,27,41 pg. 641
Example
Find each root, if possible.
a. b. c.
Solution
a.
b.
c.
4 256 5 243 4 1296
4 256
5 243
4 1296
4 because 4 4 4 4 256.
53 because ( 3) 243.
An even root of a negative number
is not a real number.
Try Q 33,35,37 pg. 641
Example
Write each expression in terms of an absolute value.
a. b. c.
Solution
a.
b.
c.
2( 5) 2( 3)x 2 6 9w w
2( 5)
2( 3)x
2 6 9w w
5 5
3x
2( 3)w 3w
Try Q 45,49,51 pg. 641
Example
If possible, evaluate f(1) and f(2) for each f(x).
a. b.
Solution
a. b.
( ) 5 1f x x 2( ) 4f x x
(1) 5(1) 1
6
f
( 2) 5( 2) 1
9 undefined
f
2(1) 1 4
5
f
2( 2) ( 2) 4
8
f
( ) 5 1f x x 2( ) 4f x x
Try Q 61,63 pg. 641
Example
Calculate the hang time for a ball that is kicked 75
feet into the air. Does the hang time double when a
ball is kicked twice as high? Use the formula
Solution
The hang time is
The hang time is
The hang times is less than double.
1( )
2T x x
1(75) 75
2T 4.3 sec
1(150) 150
2T 6.1 sec
Try Q 75,89 pg. 642
Example
Find the domain of each function. Write your answer
in interval notation.
a. b.
Solution
Solve 3 – 4x 0.
The domain is
( ) 3 4f x x 2( ) 4f x x
3 4 0
4 3
3
4
x
x
x
3, .4
b. Regardless of the value
of x; the expression is
always positive. The
function is defined for all
real numbers, and it
domain is , .
Try Q 75,89 pg. 641
Section 10.2
Rational
Exponents
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Basic Concepts
• Properties of Rational Exponents
Example
Write each expression in radical notation. Then
evaluate the expression and round to the nearest
hundredth when appropriate.
a. b. c.
Solution
a. b.
c.
1/249 1/526 1/2
6x
1/249 49 7 1/526
1/2(6 )x 6x
Try Q 37,54,59,63 pg. 650
Example
Write each expression in radical notation. Evaluate
the expression by hand when possible.
a. b.
Solution
a.
b.
2/3
8 3/410
2/3
8 2
3 8 2
2 4
3/410 34 10 4 1000
Example
Write each expression in radical notation. Evaluate
the expression by hand when possible.
a. b.
Solution
a.
3/481 4/514
/4381
43/(81)
3
4 81
Take the fourth root of
81 and then cube it.
33
27
b. 4/514 Take the fifth root of 14
and then fourth it.
54/14
4
5 14
Cannot be evaluated
by hand.
Try Q 47,51 pg. 650
Example
Write each expression in radical notation and then
evaluate.
a. b.
Solution
a. b.
1/481 2/364
1/4812/364
1/4
1
81
4
1
81
1
3
3/2
1
64
2
3
1
64
2
1
4
1
16
Try Q 53,55 pg. 650
Example
Use rational exponents to write each radical
expression.
a.
b.
c.
d.
7 3x
3
1
b3/2 b
3/7 x
25 ( 1)x2/5( 1) x
2 24 a b2 2 1/4( ) a b
Try Q 53,55 pg. 650
Example
Write each expression using rational exponents and
simplify. Write the answer with a positive exponent.
Assume that all variables are positive numbers.
a. b. 4x x34 256x1/2 1/4x x
1/2 1/4x
3/4x
1/43(256 )x
1/4 3 1/4256 ( )x
3/44x
Example (cont)
Write each expression using rational exponents and
simplify. Write the answer with a positive exponent.
Assume that all variables are positive numbers.
c. d.
5
4
32x
x
1/33
27
x
1/5
1/4
(32 )x
x
1/5 1/5
1/4
32 x
x
1/5 1/42x
1/202x
1/20
2
x
1/3
3
27
x
1/3
3 1/3
27
( )x
3
x
Try Q 77,83,91,97 pg. 650
Example
Write each expression with positive rational
exponents and simplify, if possible.
a. b.
Solution
a.
b.
4 2x
1/4
1/5
y
x
1/4
1/2( 2)x
1/8( 2)x
1/5
1/4
x
y
4 2x
1/4
1/5
y
x
Try Q 85,89,95 pg. 650
Section 10.3
Simplifying
Radical
Expressions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.
Objectives
• Product Rule for Radical Expressions
• Quotient Rule for Radical Expressions
Product Rule for Radicals
Consider the following example:
Note: the product rule only works when the radicals
have the same index.
4 25 2 5 10
4 25 100 10
Example
Multiply each radical expression.
a.
b.
c.
36 4
3 38 27
4 41 1 1 1 1
4 16 4 256 4
3 38 27 216 6
4 441 1 1
4 16 4
36 4 144 12
Try Q 13,15,21 pg. 659
Example
Multiply each radical expression.
a.
b.
c.
2 4x x
3 23 5 10a a
44 4
3 7 2121
x y xy
y x xy
3 32 3 35 10 50 50a a a a
443 7x y
y x
2 4 6 3x x x x
Try Q 23,51,57,61 pg. 659-60
Example
Simplify each expression.
a.
b.
c.
500
3 40
72
100 5 10 5
3 3 38 5 2 5
36 2 6 2
Try Q 73,75,77,79 pg. 660
Example
Simplify each expression. Assume that all variables
are positive.
a. b.
c.
449x575y
3 23 3 9a a w
4 249 7x x 4 325y y
4 325y y
25 3yy3 23 9a a w
3 327a w
3 33 27a w
33a w
Try Q 45,85,89,91 pg. 660
Example
Simplify each expression.
a. b.
37 73 5a a1/2 1/37 7
1/3 1/5a 1/2 1/37 5/67 8/15a
1/3 1/5a a
Try Q 101,103,107 pg. 660
Quotient Rule
Consider the following examples of dividing radical
expressions:
4 2 2 2
9 3 3 3
4 4 2
9 39
Example
Simplify each radical expression. Assume that all
variables are positive.
a. b. 37
275
32
x3
3
7
27
3 7
3
5
5 32
x
5
2
x
Try Q 25,27,29 pg. 659
Example
Simplify each radical expression. Assume that all
variables are positive.
a. b. 90
10
4x y
y
90
10
9
4x y
y
4x
32x
Try Q 33,39,41 pg. 659
Example
Simplify the radical expression. Assume that all
variables are positive.
4
55
32x
y
5 4
55
32x
y
5 45
55
32 x
y
5 42 x
y
Try Q 95,97 pg. 660
Example
Simplify the expression.
Solution
1 1 x x
1 1 x x ( 1)( 1) x x
2 1 x
Example
Simplify the expression.
Solution
3 2
3
5 6
2
x x
x
3 2
3
5 6
2
x x
x3
( 3)( )
)2(
2
x
x
x
3 3 x
Try Q 63,67 pg. 660
End of week 3
You again have the answers to those problems not
assigned
Practice is SOOO important in this course.
Work as much as you can with MyMathLab, the
materials in the text, and on my Webpage.
Do everything you can scrape time up for, first the
hardest topics then the easiest.
You are building a skill like typing, skiing, playing a
game, solving puzzles.