Download Sec 1.3 - 3 - Palm Beach State College

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Sec 1.3 - 1
Chapter 1
Review of the Real Number System
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Sec 1.3 - 2
1.3
Exponents, Roots, and
Order of Operations
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Sec 1.3 - 3
1.3 Exponents, Roots, and Order of Operations
Objectives
1.
2.
3.
4.
5.
Use exponents.
Identify exponents and bases.
Find square roots.
Use the order of operations.
Evaluate algebraic expressions
for given values of variables.
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Sec 1.3 - 4
1.3 Exponents, Roots, and Order of Operations
Using Exponents
Factors are two or more numbers whose product is a
third number. Exponents are a way of writing products
of repeated factors.
Exponent
81  3  3  3  3  3
4
4 factors of 3
Base
34, read as “3 to the fourth power”, uses 3 as a
factor 4 times and equals 81.
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Sec 1.3 - 5
1.3 Exponents, Roots, and Order of Operations
Using Exponents
Exponential Expression
If a is a real number and n is a natural number,
a  a  a  a  a ,
n
n factors of a
n
where n is the exponent, a is the base, and a is
an exponential expression. Exponents are also
called powers.
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Sec 1.3 - 6
1.3 Exponents, Roots, and Order of Operations
Using Exponential Notation
Write each expression
Using exponents:
Exponential notation:
5
6·6·6·6·6
6
(0.7) (0.7) (0.7) (0.7)
(0.7)
 2  2 
  9   9 



 2
 9 


40
2
3
m· m·m
m
(–y) (–y) (–y) (–y)
(–y)4
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Sec 1.3 - 7
1.3 Exponents, Roots, and Order of Operations
Evaluating Exponential Expressions
Evaluate the
expression:
7
Exponential notation:
2
7 · 7 = 49
(0.2)
 2
 5 


m
3
(0.2) (0.2) (0.2)= 0.008
3
8
 2  2  2 





 5  5  5 
125




4
(–4)
m·m·m·m
4
(–4) (–4) (–4) (–4) = 256
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Sec 1.3 - 8
1.3 Exponents, Roots, and Order of Operations
Tips to Remember
The product of an even number of negative factors is
positive.
The product of an odd number of negative factors is
negative.
To raise a number to a power on a calculator, enter the
following:
E.g., 23
2 yx 3 =
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or
2 xy 3 =
Sec 1.3 - 9
1.3 Exponents, Roots, and Order of Operations
Identifying Exponents and Bases
Identify the
Exponent and Base
11
Exponent
Base
2
2
11
3
3
4
4
–4
5
0.8
–4
(–4)
4
–(0.8)
5
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Sec 1.3 - 10
1.3 Exponents, Roots, and Order of Operations
Be Sure to Identify the Base Correctly
CAUTION
a  1 a  a  a  a
n
The base is a.
n factors of a
 a 
n
  a  a    a 
The base is  a.
n factors of  a
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Sec 1.3 - 11
1.3 Exponents, Roots, and Order of Operations
Square Roots
Squaring a number and taking its square root are
opposites.
Square
8

2
 8
2
88
Square Root
 64
  8  8   64
64  8
 64  8
64 has two square roots: 8 and –8.
Principle (positive) square root of 64 is denoted with
Negative square root of 64 is denoted with 
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.
.
Sec 1.3 - 12
1.3 Exponents, Roots, and Order of Operations
Principle Square Roots
If y is the principle (positive) square root of x, we write y = x .
This means that y must be positive, and y 2 = x.
The square of any nonzero real number is positive; so, x must
be positive.
The square root of a nonnegative number is not a real number.
36  6 since 6  36
2
36 is not a real number since
no real number squared  36
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Sec 1.3 - 13
1.3 Exponents, Roots, and Order of Operations
Finding Square Roots
Each square root is given.
49  7
since 49 is positive and 7 2  49.
0.25  0.5
since .25 is positive and  0.5   0.25.
0 0
since 02  0.
2
2
16
16
4
since
is positive and   
49
49.
7
since the negative sign is outside the
 121  11
radical.
is not a real number since the negative sign
121
is inside the radical and no real number squared
equals  121.
16 4

49 7
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Sec 1.3 - 14
1.3 Exponents, Roots, and Order of Operations
Order of Operations
When an expression involves more than one
operation symbol, use the following:
1. Work separately above and below any fraction bar.
2. If grouping symbols such as parentheses ( ), square brackets
[ ], or absolute value bars | | are present, start with the
innermost set and work outward.
3. Evaluate all powers, roots, and absolute values.
4. Do any multiplications or divisions in order, working from left to
right.
5. Do any additions or subtractions in order, working from left to
right.
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Sec 1.3 - 15
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
Simplify:
–7 + 4 • 6 = –7 + 4 • 6
= –7 + 24
=17
6 + 18 ÷ (– 3) • 2 = 6 + 18 ÷ (– 3) • 2
= 6 + (–6) • 2
= 6 + (–12)
= –6
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Sec 1.3 - 16
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
Simplify:
5 • 42 + 10 ÷ ( 8 – 6) = 5 • 42 + 10 ÷ ( 8 – 6)
= 5 • 42 + 10 ÷ 2
= 5 • 16 + 10 ÷ 2
= 80 + 10 ÷ 2
= 80 + 5
= 85
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Sec 1.3 - 17
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
Simplify:
1 • 12 + (– 18 + 15 ÷ 3)
1 • 12 + (– 18 + 15 ÷ 3)
=
3
3
= 1 • 12 + (– 18 + 5)
3
= 1 • 12 + (– 13)
3
= 4 + (– 13)
= –9
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Sec 1.3 - 18
1.3 Exponents, Roots, and Order of Operations
Using Order of Operations
Simplify:
3
3 7
3 7

2 8  4 9
2 8  4 9
27  7

2 8  4 3
3
27  7

1 6  12
20

4
5
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Sec 1.3 - 19
1.3 Exponents, Roots, and Order of Operations
Algebraic Expressions
Any collection of numbers, variables, operation
symbols, and grouping symbols, such as
3xy,  4b  6c,
and
 x 2 y  7
2
is called an algebraic expression. Algebraic
expressions have different numerical values for
different values of the variables.
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Sec 1.3 - 20
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions for Given Values of Variables
The cost for a season pass to a state park is $12
per person. The amount of dollars a family of x
members would pay can be represented by $12x.
Cost per person = $12
Total cost = $12x
Number of persons = x
3 member family
5 member family
Total cost = $12x
Total cost = $12x
12 • 3 = $36
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12 • 5 = $60
Sec 1.3 - 21
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
If c = 4 and b = –3, evaluate the expression:
3c – 7b = 3(4) – 7(–3)
= 12 + 21
= 33
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Sec 1.3 - 22
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
Use parentheses
to avoid errors.
Given
r = –1
s = 64
t = –7
Substitute and
evaluate.
2
t2  r
7    1


3 s
3  64
49   1

38
49  1

38
50

5
 10
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Sec 1.3 - 23
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
The price per gallon of gasoline can be
approximated for the years 2006 – 2008 by
substituting a given year for x in the expression
0.17 x – 338.07
and then evaluating. Approximate the price of a
gallon of gas in the year 2007.
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Sec 1.3 - 24
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
The approximate price of a gallon of gas in the
year 2007, rounded to the nearest cent is
0.17x – 338.07
= 0.17(2007) – 338.07
= 341.19 – 338.07
= $3.12
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Sec 1.3 - 25
1.3 Exponents, Roots, and Order of Operations
Evaluating Expressions
We can create a table to show how the price of
gas changed during these years.
Year
Price Per Gallon
2006
$2.95
2007
$3.12
2008
$3.29
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Sec 1.3 - 26