Date post: | 23-Dec-2015 |
Category: |
Documents |
Upload: | solomon-thompson |
View: | 222 times |
Download: | 0 times |
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.3 - 2
Review of the Real Number System
Chapter 1
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.3 - 3
1.3
Exponents, Roots, and
Order of Operations
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 4
1.3 Exponents, Roots, and Order of Operations
Objectives
1. Use exponents.
2. Identify exponents and bases.
3. Find square roots.
4. Use the order of operations.
5. Evaluate algebraic expressionsfor given values of variables.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 5
Using Exponents
Factors are two or more numbers whose product is a
third number. Exponents are a way of writing products
of repeated factors.
1.3 Exponents, Roots, and Order of Operations
factors of
4
4 3
38 3 31 3 3
Base
Exponent
34, read as “3 to the fourth power”, uses 3 as a factor 4 times and equals 81.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 6
Using Exponents
Exponential Expression
If a is a real number and n is a natural number,
1.3 Exponents, Roots, and Order of Operations
factors of
, a
n
n
a a a a a
where n is the exponent, a is the base, and is an exponential expression. Exponents are also called powers.
na
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 7
1.3 Exponents, Roots, and Order of Operations
Using Exponential Notation
Write each expression
Using exponents:Exponential notation:
6 · 6 · 6 · 6 · 6 65
(0.7) (0.7) (0.7) (0.7) (0.7)40
m · m · m m3
(–y) (–y) (–y) (–y) (–y)4
2 2
9 9
22
9
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 8
1.3 Exponents, Roots, and Order of Operations
Evaluating Exponential Expressions
Evaluate the expression:
Exponential notation:
72 7 · 7 = 49
(0.2)3 (0.2) (0.2) (0.2)= 0.008
m4 m · m · m · m
(–4)4 (–4) (–4) (–4) (–4) = 256
32
5
2 2 2 8
5 5 5 125
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 9
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 = or 2 xy 3 =
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 10
1.3 Exponents, Roots, and Order of Operations
Identifying Exponents and Bases
Identify the Exponent and Base
Exponent Base
112 2 11
–43 3 4
(–4)4
4 –4
–(0.8)5 5 0.8
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 11
1.3 Exponents, Roots, and Order of Operations
Be Sure to Identify the Base Correctly
CAUTION
factors of
factors of
The base is 1 .
. The base is
n
n a
n
n a
a a a a a a
a a a a a
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 12
Square Roots
1.3 Exponents, Roots, and Order of Operations
Squaring a number and taking its square root are opposites.
2
2
Square Square Root
8 64 64
64 64
8 8 8
8 8 8 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
.
.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 13
Principle Square Roots
1.3 Exponents, Roots, and Order of Operations
2
If is the principle (positive) square root of , we write .
This means that must be positive, and .
The square of any nonzero real number is positive; so, must
be positive.
The square roo t o
=
=
y x
x
x
y
y
x
y
f a is nnonne ot a gative num real number ber.
2
not real numb
6 since 6
is a since
no real number s
e
quared
36 36
36
36
r
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 14
Finding Square Roots
1.3 Exponents, Roots, and Order of Operations
Each square root is given.
2
2
2
2
since is positive and .
0 since is positive and 0 .
since .
since is positive and
since the negative sign is outside the
radic
49 49 49
0.25 .25 .25
0 0
16 16 16
49 4
7 7
.5 0.5
0 0
4 4
7 9 497 .
121al.
11
no real number squared
equals
is not a real number since the negative sign
is inside the radical and
.
1
21
121
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 15
Order of Operations
1.3 Exponents, Roots, and 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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 16
= 6 + 18 ÷ (– 3) • 2
–7 + 4 • 6 = –7 + 4 • 6
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
Simplify:
6 + 18 ÷ (– 3) • 2
= –7 + 24
=17
= 6 + (–6) • 2
= 6 + (–12)
= –6
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 17
5 • 42 + 10 ÷ ( 8 – 6)
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
Simplify:
= 5 • 42 + 10 ÷ ( 8 – 6)
= 5 • 42 + 10 ÷ 2
= 5 • 16 + 10 ÷ 2
= 80 + 10 ÷ 2
= 80 + 5
= 85
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 18
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
Simplify:
13
• 12 + (– 18 + 15 ÷ 3) 13
• 12 + (– 18 + 15 ÷ 3)=
13
• 12 + (– 18 + 5)=
13
• 12 + (– 13)=
4 + (– 13)=
–9=
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 19
Using Order of Operations
1.3 Exponents, Roots, and Order of Operations
Simplify: 33 7
2 8 4 9
3 7
2 8
3
94
5
8
27
3
7
2 4
1 1
7
2
2 7
6
20
4
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 20
Algebraic Expressions
1.3 Exponents, Roots, and Order of Operations
Any collection of numbers, variables, operation symbols, and grouping symbols, such as
26 2 7, ,3 4 xy b c x y
is called an algebraic expression. Algebraic expressions have different numerical values for different values of the variables.
and
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 21
Evaluating Expressions for Given Values of Variables
1.3 Exponents, Roots, and Order of Operations
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
Number of persons = xTotal cost = $12x
3 member family 5 member family
Total cost = $12x Total cost = $12x
12 • 3 = $36 12 • 5 = $60
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 22
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
If c = 4 and b = –3, evaluate the expression:
3c – 7b = 3(4) – 7(–3)
= 12 + 21
= 33
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 23
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
2
3
7 1
64
r = –1
s = 64
t = –7
Use parentheses to avoid errors.
Given
Substitute and evaluate.
2
3
t r
s
10
149
83
49
3 8
1
50
5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 24
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 25
Evaluating Expressions
1.3 Exponents, Roots, and Order of Operations
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