Download bei05_ppt_01

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 1
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Introduction to Algebraic
Expressions
Introduction to Algebra
The Commutative, Associative, and
Distributive Laws
Fraction Notation
Positive and Negative Real Numbers
Addition of Real Numbers
Subtraction of Real Numbers
Multiplication and Division of Real Numbers
Exponential Notation and Order of Operations
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1.1
Introduction to Algebra

Algebraic Expressions

Translating to Algebraic Expressions

Translating to Equations
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Algebraic Expression

An algebraic expression consists of
variables and/or numerals often with
operation signs and grouping symbols.
Examples:
y  75
4a  9b
36
y
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
5a ( x  w)
Slide 1- 4
Example
Evaluate each expression for the given values.
a)
b)
a + b for
a = 21 and b = 74
7xy for
x = 3 and y = 6
Solution
a + b = 21 + 74
= 95
Solution
7xy = 7 • 3 • 6
= 126
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 5
Example
The area of a triangle
with a base length b
and height of length h
is given by the formula
A = ½ bh. Find the
area when b is 12 m
and h is 7.2 m.
Solution
1
A  bh
2
1
 (12m)(7.2m)
2

1
(12)(7.2)(m)(m)
2
 6(7.2)m2
 43.2m 2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 6
Translating to Algebraic Expressions
Addition
Subtraction
Multiplication
Division
added to
subtracted from
multiplied by
divided by
sum of
difference of
product of
quotient of
plus
minus
times
divided into
more than
less than
twice
ratio of
increased by
decreased by
of
per
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 7
Example
Translate each phrase to an algebraic expression.
a) 9 more than y
b) 7 less than x
c) the product of 3 and twice w
Solution
Phrase
Algebraic Expression
a) 9 more than y
y+9
b) 7 less than x
x7
c) the product of 3 and twice w
3•2w or 2w • 3
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 8
Translating to Equations


The symbol = (“equals”) indicates that the
expressions on either side of the equals
sign represent the same number.
An equation is a number sentence with
the verb =.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 9
Solution
A replacement or substitution that makes
an equation true is called a solution. Some
equations have more than one solution, and
some have no solution. When all solutions
have been found, we have solved the
equation.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 10
Example
Determine whether 12 is a solution of
x + 4 = 16.
Solution
x + 4 = 16
12 + 4 | 16
16 = 16
Writing the equation
Substituting 12 for x
16 = 16 is TRUE.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 11
Example
Translate the problem to an equation.
What number added to 127 is 403?
Solution
Let y represent the unknown number.
What number added to 127 is 403?

y





127  403
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 12
1.2
The Commutative, Associative,
and Distributive Laws

Equivalent Expressions

The Commutative Laws

The Associative Laws

The Distributive Law

The Distributive Law and Factoring
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Commutative Laws
For Addition. For any numbers a and b,
a + b = b + a.
(Changing the order of addition does not affect the
answer.)
For Multiplication. For any numbers
a and b,
ab = ba.
(Changing the order of multiplication does not
affect the answer.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 14
Example
Use the commutative laws to write an
expression equivalent to each of the following.
a) r + 7
b) 12y
c) 9 + st
Solution
a) r + 7 is equivalent to 7 + r
b) 12y is equivalent to y • 12
c) 9 + st is equivalent to st + 9
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 15
The Associative Laws
For Addition. For any numbers a, b, and c,
a + (b + c) = (a + b) + c.
(Numbers can be grouped in any manner for
addition.)
For Multiplication. For any numbers
a, b, and c,
a • (b • c) = (a • b) • c.
(Numbers can be grouped in any manner for
multiplication.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 16
Example
Use the associative laws to write an
expression equivalent to each of the
following.
a) t + (4 + y)
b) (12y)z
Solution
a) t + (4 + y) is equivalent to (t + 4) + y
b) (12y)z is equivalent to 12(yz)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 17
The Distributive Law
For any numbers a, b, and c,
a(b + c) = ab + ac.
(The product of a number and a sum can be
written as the sum of two products.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 18
Example
Multiply: 4(x + 7)
Solution
4( x  7)  4 x  4 7
 4x  28
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 19
Example
Multiply: 7(x + y + 4z)
Solution
7( x  y  4 z )  7 x  7 y  7 4 z
 7 x  7 y  (7 4) z
 7 x  7 y  28 z
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 20
Example
Multiply: (a + 3)2
Solution
(a  3)2  a 2  3 2
 2a  6
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 21
The Distributive Law and Factoring

If we use the distributive law in reverse, we have the
basis of a process called factoring.

To factor an expression means to write an
equivalent expression that is a product.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 22
Example
Use the distributive law to factor each of the
following.
a) 5x + 5y
b) 8y + 32w + 8
Solution
a) 5x + 5y
5x + 5y = 5(x + y)
b) 8y + 32w + 8 = 8y + 84w + 81
= 8(y + 4w + 1)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 23
1.3
Fraction Notation

Factors and Prime Factorizations

Fraction Notation

Multiplication, Division, and Simplification

More Simplifying

Addition and Subtraction
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Factors and Prime Factorizations

Natural Numbers can be thought of as the counting
numbers: 1, 2, 3, 4, 5…
(The dots indicated that the established pattern
continues without ending.)
To factor a number, we simply express it as a
product of two or more numbers.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 25
Example
Write several factorizations of 18. Then list all
the factors of 18.
Solution
118,
2 9,
3 6, 2 3 3
The factors of 18 are: 1, 2, 3, 6, 9, and 18.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 26
Prime Number
A prime number is a natural
number that has exactly two
different factors: the number itself
and 1. The first several primes are
2, 3, 5, 7, 11, 13, 17, 19, and 23.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 27
Definitions

If a natural number, other than 1, is not prime,
we call it composite.

Every composite number can be factored into
a product of prime numbers. Such a
factorization is called the prime
factorization of that composite number.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 28
Example
Find the prime factorization of 48.
Solution
48  6 8
2 3 4 2
2 3 2 2 2
The prime factorization of 48 is
2 2 2 2 3.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 29
Fraction Notation

The top number is called the numerator and the
bottom number is called the denominator.
5  Numerator
7  Denominator
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 30
Fraction Notation for 1
For any number a, except 0,
a
 1.
a
(Any nonzero number divided by
itself is 1.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 31
Multiplication of Fractions
For any two fractions a/b and c/d,
a c ac

.
b d bd
(The numerator of the product is the
product of the two numerators . The
denominator of the product is the product
of the two denominators.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 32
Example
5
3
2
3
Multiply: a)
b)
w y
7 5
Solution
a)
b)
2 3 23 6


7 5 7 5 35
5 3 5 3 15


w y w y wy
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 33

Two numbers whose product is 1 are
reciprocals, or multiplicative inverses.
Example
3
4
The reciprocal of 4 is 3 because
3 4 12

 1.
4 3 12
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 34
Division of Fractions
To divide two fractions, multiply by the
reciprocal of the divisor:
a c a d
 
.
b d b c
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 35
Example
1
2
Divide: 
4 3
Solution
1 2 1 3
 
4 3 4 2
3

8
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 36
The Identity Property of 1
For any number a,
a ● 1 = 1 ● a = a.
(Multiplying a number by 1 gives the same
number.) The number 1 is called the
multiplicative identity.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 37
Example
Simplify:
20
35
Solution
20 4 5

35 7 5
Factoring the numerator and the
denominator using a common factor of 5.
4 5

7 5
Rewriting as a product of two fractions
4
4
 1
7
7
Using the identity property of 1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 38
Addition and Subtraction of
Fractions
For any two fractions a/d and b/d,
a b a b
a b ab
 
.
 
and
d d
d
d d
d
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 39
Example
Add and simplify:
Solution
7
9

12 20
7 9
7 5 9 3



12 20 12 5 20 3
Using 60 as the common
denominator
35 27


60 60
Perform the multiplication
62 31


60 30
Adding fractions &
simplifying
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 40
Example
Perform the indicated operation and, if possible,
simplify.
3
8
Solution
8
14
3 83

14 14
2223

27
2 223

2 7
Removing a factor equal to 1
12

7
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 41
Example
Perform the indicated operation and, if possible,
simplify.
Solution
3
7
9
5
3 9
 
7 5




3 5
7 9
35
733
3 5
7 3 3
5
21
Removing a factor equal to 1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 42
1.4
Positive and Negative
Real Numbers

The Integers

The Rational Numbers

Real Numbers and Order

Absolute Value
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

A set is a collection of objects.

The integers consist of all whole numbers and their
opposites.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 44
Set of Integers
The set of integers =
{…−4, −3, −2, −1, 0, 1, 2, 3, 4, …}
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 45
Example
State which integer(s) corresponds to the situation.
The lowest point in Australia is Lake Eyre at 15 m
below sea level and the highest point is Mt. Kosciuszko
at 2229 m above sea level.
Solution
The integer −15 corresponds to 15 m below sea level.
The integer 2229 corresponds to 2229 m above sea level.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 46
Set of Rational Numbers
The set of rational numbers =
a

 | a and b are integers and b  0  .
b

This is read “the set of all numbers a
over b, where a and b are integers and b
does not equal zero.”
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 47
Example
Convert to decimal notation:
Solution
3
.
8
0.375
8 3.000
24
60
56
40
40
0
← The remainder is 0.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 48
Example
Convert to decimal notation: 5 .
22
Solution
0.2272
22 5.000
44
60
44
160
5
 0.227
22
154
60
44
16
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 49
Set of Real Numbers
The set of real numbers = The set of all
numbers corresponding to points on the
number line.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 50
Example
Example
Which numbers in the following list are (a) whole numbers? (b) integers? (c)
rational numbers? (d) irrational numbers? (e) real numbers?
9
27,
, 0, 4, 65,
5
40, 72
Solution
a) whole numbers: 0, 4, 65, 72
b) integers: −27, 0, 4, 65, 72
c) rational numbers: 27, 9 , 0, 4, 65, 72
5
d) irrational numbers:
40
e) real numbers: 27,
9
, 0, 4, 65,
5
40, 72
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 51

Real numbers are named in order on the number
line, with larger numbers further to the right.
< mean “less than”

> means “greater than”

−5 < 8
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−2 > −9
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 52
Example
Use either < or > for the blank to write a true
statement. (a) −3.42 ____ 2.35
(b) 7 ____ −15
(c) −11 ____ −9
Solution
a) Since −3.42 is to the left of 2.35, we have
−3.42 < 2.35.
b) Since 7 is to the right of −15, we have 7 > −15.
c) Since −11 is to the left of −9, we have −11 < −9.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 53
More Inequalities
≤
means “is less than or equal to”

means “is greater than or equal to”
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 54
Example
Classify each inequality as true or false.
a) −9 ≤ 7
b) −8  −8
c) −7  −2
Solution
a) −9 ≤ 7 is true because −9 < 7.
b) −8  −8 is true because −8 = −8.
c) −7  −2 is false since −7 > −2 nor −7 = −2 is
true.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 55
Absolute Value
We write |a|, read “the absolute value of a,” to
represent the number of units that a is from
zero.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 56
Example
Find the absolute value:
a) |–4|
b) |3.8|
c) |0|
Solution
a) |–4| = 4 since –4 is 4 units from 0.
b) |3.8| = 3.8 since 3.8 is 3.8 units from 0.
c) |0| = 0 since 0 is 0 units from itself.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 57
1.5
Addition of Real Numbers

Adding with the Number Line

Adding Without the Number Line

Problem Solving

Combining Like Terms
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Adding with a Number Line
To add a + b on a number line, we start at a and
move according to b.
a) If b is positive, we move to the right
(the positive direction).
b) If b is negative, we move to the left (the
negative direction).
c) If b is 0, we stay at a.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 59
Example
Add: −3 + 7.
Solution
Locate the first number −3, and then move 7 units to
the right
Start at −3
Move 7 units to the right.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−3 + 7 = 4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 60
Example
Add: −2 + (−3).
Solution
After locating −2, we move 3 units to the left.
Move
Start at −2
3 units to the left.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−2 + (−3) = −5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 61
Rules for Addition of Real Numbers
1. Positive numbers: Add as usual. The answer is positive.
2. Negative numbers: Add the absolute values and make the
answer negative.
3. A positive and a negative number: Subtract the smaller
absolute value from the greater absolute value.
a) If the positive number has the greater absolute value,
the answer is positive.
b) If the negative number has the greater absolute value,
the answer is negative.
c) If the numbers have the same absolute value, the
answer is 0.
4. One number is zero: The sum is the other number.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 62
Adding Without the Number Line
a) −9 + (−11)
b) −34 + 15
c) −2.3 + 7.4
d) 2.4 + (−2.4)
Two negatives, add the
absolute value, answer is −20.
A negative and a positive,
subtract and the answer is
negative, −19.
A negative and a positive,
subtract and the answer is
positive, 5.1.
A negative and a positive,
subtract and the answer is 0.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 63
Example
Add: 17 + (−3) + 9 + 16 + (−4) + (−12).
Solution
17 + (−3) + 9 + 16 + (−4) + (−12)
= 17 + 9 + 16 + (−3)+ (−4) + (−12) Using the commutative law
= (17 + 9 + 16) +[(−3)+ (−4) + (−12)] Using the associative law
= 42 + (−19)
= 23
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 64
Problem Solving
During the first two weeks of the semester, 6 students withdrew
from Mr. Lange’s algebra class, 9 students were added to the
class, and 4 students were dropped as “no-shows.” By how
many students did the original class size change?
The 1st plus the 2nd plus the 3rd is the total
change
6
 change
 change = change


9
(4)
 Total change
The original class size dropped by one.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 65
Combining Like Terms

When two terms have variable factors that are
exactly the same, the terms are called like or similar
terms.
Example
Combine like terms −5x + 7x.
Solution −5x + 7x = (−5 + 7)x
= 2x
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 66
Example
Combine like terms: 3a + (−4b) + (−8a) + 10b
Solution
3a + (−4b) + (−8a) + 10b
= 3a + (−8a) + (−4b) + 10b
= (3 +(−8))a + (−4 + 10)b
= −5a + 6b
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 67
1.6 Subtraction of Real Numbers

Opposites and Additive Inverses

Subtraction

Problem Solving
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Opposite
The opposite, or additive inverse, of a
number a is written −a (read “the opposite of
a” or “the additive inverse of a”).
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 69
The Opposite of an Opposite
For any real number a,
−(−a) = a.
(The opposite of the opposite of a is a.)
Example
Find −x and −(−x) when x = 27.
Solution
If x = 27, then −x = −27.
If x = 27, then −(−27) = 27.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 70
The Law of Opposites
For any two numbers a and –a,
a + (–a) = 0.
(When opposites are added, their sum is 0.)
The opposite of 4 is –4.
The opposite of 0 is 0.
The opposite of –15 is 15.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 71
Subtraction of Real Numbers
For any real numbers a and b,
a – b = a + (–b)
(To subtract, add the opposite, or additive
inverse, of the number being subtracted.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 72
Example
Subtract each of the following.
a) 3  7
b) 8 – (–6)
c) –3.4 – (–2.6)
1  4
d)    
7  7
Solution
a) 3 – 7 = 3 + (−7) = −4
b) 8 – (–6) = 8 + 6 = 14
c) –3.4 – (–2.6) = –3.4 + 2.6
= –0.8
d) 1    4   1  4  5
7  7  7 7 7
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 73
Example
Simplify: 9 − (−3) − 5 − (−8) + 4.
Solution
9 − (−3) − 5 − (−8) + 4 = 9 + 3 + (−5) + 8 + 4
= 19
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 74
Example
Combine like terms.
a) 4 + 5x − 19x
b) −7a − 4b − 3a + 12b
Solution
a) 4 + 5x − 19x = 4 + 5x + (−19x)
= 4 + (5 + (− 19))x
= 4 + (− 14)x
= 4 − 14x
b) −7a − 4b − 3a + 12b = −7a + (−4b) + (−3a) + 12b
= −7a + (−3a) + (−4b) + 12b
= −10a + 8b
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 75
Problem Solving Example
The lowest point in Australia is Lake Eyre at 15 m
below sea level and the highest point is Mt.
Kosciuszko at 2229 m above sea level. What is the
difference in elevation?
Solution
Higher elevation − Lower elevation
= 2229 − (−15)
= 2244 m
The difference in elevation is 2244 m.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 76
1.7
Multiplication and Division
of Real Numbers

Multiplication

Division
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Product of a Negative
Number and a Positive Number
To multiply a positive number and a
negative number, multiply their absolute
values. The answer is negative.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 78
Example
Multiply:
a) 9(−3) = −27
b)−4(12) = −48
c) 3
4
 3
9





20
 5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 79
The Multiplicative Property
of Zero
For any real number a,
0  a = a  0 = 0.
(The product of 0 and any real number is 0.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 80
Example
Multiply: 125(−721)(0)
= 125 [(−721)(0)]
= 125(0)
Using the associative law
Using the multiplicative property of 0
=0
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 81
The Product of Two Negative
Numbers
To multiply two negative numbers, multiply
their absolute values. The answer is positive.
Example
a) −9(−9) = 9 · 9 = 81
b) −4(−12) = 4 · 12 = 48
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 82
When three or more numbers are multiplied, we can order
the numbers as we please, because of the commutative and
associative laws.
Example
Multiply a) −4(−5)(−6)
b) −5(−6)(−2)(−3)
Solution
a) −4(−5)(−6) = 20(−6)
= −120
b) −5(−6)(−2)(−3) = 30  6
= 180
The product of an even number of negative numbers is positive.
 The product of an odd number of negative numbers is negative.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 83
Example
Divide, if possible.
36
a) 21 ÷ (−3)
b)
4
c)
41
0
Solution
a) 21 ÷ (−3) = −7
36
9
b)
4
c)
41
 undefined
0
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 84
Rules for Multiplication and
Division
To multiply or divide two nonzero real numbers:
1. Using the absolute values, multiply or divide,
as indicated.
2. If the signs are the same, the answer is positive.
3. If the signs are different, the answer is negative.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 85
Example
a. Divide:
b. Divide:
4  3
  
7  2
Solution
4

7
14.4
4
Solution
 2 8
  
 3  21
14.4
 3.6
4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 86
Division Involving Zero
For any real number a,
a
is undefined,
0
and for a ≠ 0,
0
 0.
a
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 87
1.8
Exponential Notation and
Order of Operations

Exponential Notation

Order of Operations

Simplifying and the Distributive Law

The Opposite of a Sum
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponential Notation
4 4 4 4 4 we write as 4
5
5 factors

The 5 is called an exponent.

The 4 is the base.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 89
Example
Write exponential notation for 777777.
Solution
Exponential notation is 76
6 is the
exponent.
7 is the base.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 90
Example
Simplify:
a) 82
b) (−8)3
c) (4y)3
Solution
a)
82 = 8  8 = 64
b) (−8)3 = (−8) (−8) (−8)
= 64(−8)
= − 512
c)
(4y)3 = (4y) (4y) (4y)
= 4  4 4 y y y
= 64y3
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 91
Exponential Notation
For any natural number n,
n factors
n
b means b b b b ... b.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 92
Rules for Order of Operations
1. Calculate within the innermost grouping
symbols, ( ), [ ], { }, | |, and above or
below fraction bars.
2. Simplify all exponential expressions.
3. Perform all multiplication and division
working from left to right.
4. Perform all addition and subtraction
working from left to right.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 93
Example
Simplify:
20  4  2  7
Solution
20  4  2  7  20  8  7
 12  7
 19
Multiplying
Subtracting and adding
from left to right
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 94
Example
Evaluate 16  8(7  y)2 for y = 2.
Solution
16  8(7  y)2 = 16  8(7  2)2
= 16  8(5)2
= 16  8(25)
= 2(25)
= 50
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 95
Example
Simplify: 20  5  42 [(13  4)  8]  23.
Solution
20  5  42 [(13  4)  8]  23  4  16[17  8]  8
 4  16(9)  8
 4  144  8
 140
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 96
Example
Simplify:
Solution
4(6  2)  8(8  3)
6(4  2)  22
4(6  2)  8(8  3) 4(8)  8(5)

2
6(2)  4
6(4  2)  2
32  40

12  4
72

9
8
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 97
Simplifying and the Distributive Law
Example
Simplify: 7 x  3  4(2 x  5)
Solution
7 x  3  4(2 x  5)  7 x  3  8 x  20 Distributive Law
 15x  23
Combining Like Terms
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 98
The Property of −1
For any real number a,
−1  a = −a.
(Negative one times a is the opposite of a.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 99
Example
Write an expression equivalent to
(4x + 3y + 5) without using parentheses.
Solution
(4x + 3y + 5) = 1(4x + 3y + 5)
= 1(4x) + 1(3y) + 1(5)
= 4x  3y  5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 100
The Opposite of a Sum
For any real numbers a and b,
−(a + b) = −a + (−b) = −a − b
(The opposite of a sum is the sum of the opposites.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 101
Example
Simplify: 8 y 2  3 y  (2 y 2  6 y)
Solution
8 y  3 y  (2 y  6 y )  8 y  3 y  2 y  6 y
2
2
2
2
 6 y2  3y
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 102
Example
Simplify: 7 w3  5  [3( w3  2)  1]
Solution
7w3  5  [3( w3  2)  1]  7 w3  5  [3w3  6  1]
 7w3  5  [3w3  5]
 7 w  5  3w  5
3
3
 4w  10
3
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1- 103