Section 6.2
PRE-ACTIVITY
PREPARATION
Exponents
Exponents enable you to simplify the presentation of a numerical expression
containing repeated multiplication into a concise form that is easier to read and
use in practical applications. For instance, you may have already used exponents
to present the prime factorization of a number.
Exponents are often utilized in the analysis of statistical data, and they are used
in science and engineering when presenting very large or very small numbers in
scientific notation. In business applications, using exponents greatly simplifies
the processes of calculating growth of savings or retirement of debt.
Although exponents can be negative as well as positive numbers, this chapter
will only address the use of positive exponents.
LEARNING OBJECTIVES
•
Write expressions in exponential notation.
•
Evaluate expressions having positive exponents.
•
Order a mixed set of signed numbers and numerical expressions.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
base
cubed
evaluate
exponential expression
exponent
raised to
exponent form
squared
exponential notation
expression
power
simplify
term
569
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
570
BUILDING MATHEMATICAL LANGUAGE
The notation you can use to represent repeated multiplication of the same number is called exponential
notation.
exponent
For example, 7 × 7 × 7 × 7 × 7 in exponent form is 75 “seven to the fifth power” or
“seven raised to the fifth power”
base
The exponent 5 denotes five factors of the base number 7.
For example, 3 • 3 • 3 • 5 • 5 written in exponential notation is 33 • 52
The base can be any kind of number—a whole number, a fraction, a decimal number, positive or
negative.
Sometimes the base number might be written within parentheses.
For example, 53 might also be written as (5)3, or 0.062 might be written as (0.06)2.
Negative number bases and fraction bases, however, must be written in parentheses for clear presentation
2
⎛ 2⎞
of the base number: (–3)5 and ⎜⎜ ⎟⎟ , for example.
⎜⎝ 5 ⎟⎠
Any base number raised to the first power means that number is a factor one time.
For example, 71 = 7,
231 =23, and so on.
Therefore, it is customary to write 71 as 7, 231 as 23, and so on.
A base number raised to the second power is commonly referred to as that number squared.
For example, 7 × 7 = 72 “seven squared”
A base number raised to the third power is often referred to as that number cubed.
For example, 21 × 21 × 21 = 213 “twenty-one cubed”
Any number to the power of zero is equal to 1.
For example, 70 = 1, 230 =1, and so on.
Terms written with exponents can be combined with other mathematical symbols to form new
expressions, as in the following examples:
7 + 32
102 + 24
Section 6.2 — Exponents
571
Expressions having exponents are sometimes referred to as exponential expressions.
To evaluate or to simplify an exponential expression is to simplify the expression to one number.
The most accurate way to simplify an exponential expression is to write out its factors and then multiply
them.
(5)3 = 5 × 5 × 5
= 25 × 5
= 125
MODELS
Model 1
Evaluate each of the following:
A
►
B
►
(0.06)2
0.06
×0.06
(0.06)2 = 0.06 × 0.06 = 0.0036 Answer
0.0036
(–3)5
(–3)5 = −3 × (–3) × (–3) × (–3) × (–3)
=
=
+9
× (–3) × (–3) × (–3)
−27
=
× (–3) × (–3)
+81
=
C
►
× (–3)
− 243
Use the shortcut for multiplying more
than two signed numbers. With five
negative factors, the answer will be
negative. Multiply 3×3×3×3×3 and
attach a negative sign:
Answer: –243
2
⎛ 2 ⎞⎟
⎜⎜ ⎟
⎜⎝ 5 ⎟⎠
2
⎛ 2 ⎞⎟
⎜⎜ ⎟ = 2 × 2 = 4 Answer
⎜⎝ 5 ⎟⎠
5 5 25
D
►
OR
102 + 24
Evaluate each term separately:
102 = 10 × 10 = 100
24 = 2 × 2 × 2 × 2 = 16
102 + 24 = 100 + 16 = 116 Answer
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
572
E
►
Evaluate (−4)2 and −42.
There is an important distinction to be made between (–4)2 and –42.
(−4)2
Because of the parentheses around the −4, the term (–4)2 represents
the base number –4 raised to the second power:
(−4) × (−4) = +16
That is, (−4)2 = +16 Answer
−42
On the other hand, the negative sign in the expression –42, indicates
“the opposite of” the base number 4 raised to the second power:
–42 = the opposite of 42
= the opposite of 4 × 4 or the opposite of 16
That is, –42 = –16 Answer
F
►
Evaluate (−4)3 and −43.
Note that, with an odd power, both simplified answers are the same.
(−4)3 = (−4) × (−4) × (−4) = −64 Answer
−43 = the opposite of 4 × 4 × 4, or the opposite of 64, or −64 Answer
Model 2
Combine your knowledge of signed numbers and exponents, and list the following sets of numerical
expressions in order from lowest to highest value.
A
►
–2, –42, 12, –10, (–5)2
Original list: –2, –42, 12, –10, (–5)2
Evaluate the expressions
with exponents:
–42 = −(4 × 4) = −16
(–5)2 = (–5) × (–5) = +25
List after evaluating
terms with exponents:
–2, –16, 12, –10,
25
Visualize on a number line (optional).
–16 –10
12
–2
–16 < –10 < –2 < 12 < 25
That is, from lowest to highest:
–42, –10, –2, 12, (–5)2
Answer
25
Section 6.2 — Exponents
B
►
32 , −
1
, 0,
2
573
2
(2.1)
, − 4.2
Original list: 32 , −
Evaluate the expressions
with exponents:
1
, 0,
2
2
(2.1)
, − 4.2
32 = 3 × 3 = 9
(2.1)2 = 2.1 × 2.1 = 4.41
List after evaluating
terms with exponents: 9,
Since two are decimals, compare as decimals.
−
1
, 0, 4.41, − 4.2
2
9, − 0.5, 0, 4.41, − 4.2
Visualize on a number line (optional).
–4.2 –0.5 0
4.41
9
–4.2 < –0.5 < 0 < 4.41 < 9
Using the original numbers,
the list from lowest to highest is:
-4.2, -
1
2
, 0, (2.1) , 32 Answer
2
ADDRESSING COMMON ERRORS
Issue
Misinterpreting
the exponent as
a factor
Incorrect
Process
Evaluate: 84
84 = 4 × 8 =32
nswer
er: 32
Answer:
Resolution
The exponent indicates
how many times the base
should be multiplied by
itself.
Write out the factors and
count them to assure that
you have written the base
as a factor as many times
as the exponent indicates.
Correct
Process
Evaluate: 84
84 = 8 × 8 × 8 × 8
= 64 × 8 × 8
= 512 × 8
= 4096
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
574
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with exponents
how to write in exponential notation
how to evaluate exponential expressions
how to order a mixed set of numbers and numerical expressions
Section 6.2
ACTIVITY
Exponents
PERFORMANCE CRITERIA
•
Writing expressions in exponent form
– correct notation
•
Evaluating expressions with exponents
– accuracy
•
Ordering a set of numbers and numerical expressions
– accurate conversions to the same form
– correct order
CRITICAL THINKING QUESTIONS
1. What is the meaning of the exponent?
2. What kind of number(s) can you use for the base?
3. What does it mean to have an exponent of one (1)?
575
576
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
4. What is the value of one (1) raised to any power?
5. What is the result when you evaluate a negative number raised to an even number exponent?
6. What is the result when you evaluate a negative number raised to an odd number exponent?
7. How can you represent a number added to itself four times versus a number multiplied by itself four
times?
Section 6.2 — Exponents
TIPS
FOR
577
SUCCESS
• Write out the factors when evaluating an expression with exponents.
• Use the shortcut for determining the sign of the simplified answer when the base of the exponential
expression is a negative number:
—An even exponent means an even number of negative factors, resulting in a positive answer.
—An odd exponent means an odd number of negative factors, resulting in a negative answer.
• When the base is a decimal number, pay careful attention to the placement of the decimal point in the
simplified answer.
DEMONSTRATE YOUR UNDERSTANDING
1. Write each expression in exponent form and then evaluate it.
Expression
a) 7 × 7 × 3 × 3 × 3
b) 2 • 2 • 2 • 2 • 5 • 5 • 11
c) 3 × 3 + 5 × 5 × 5
Exponent Form
Evaluation
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
578
2. Evaluate the following exponential expressions:
a)
43 × 22 × 30
b)
⎛1 ⎞⎟
⎜⎜ ⎟
⎜⎝5 ⎟⎠
c)
⎛ 2 ⎞⎟
⎜⎜− ⎟
⎝⎜ 3 ⎟⎠
d)
–112
e)
52 + 103
f)
(0.03)3
g)
202
h)
157
3
2
3. Which is greater? 23 or 32
4. Which is greater? 53 or 35
Section 6.2 — Exponents
579
5. What number is 1 less than 103?
6. Order the following from lowest to highest value:
IDENTIFY
AND
CORRECT
THE
1
, − 52 , 12, − 10,
8
2
(−5)
ERRORS
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column.
Worked Solution
What is Wrong Here?
1) Evaluate: 62 • 25
Identify the Errors
Correct Process
62 × 25
Used the exponent as a factor.
62 means 6 × 6,
not 2 × 6.
25 means 2 × 2 × 2 × 2 × 2,
not 5 × 2
=6×6×2×2×2×2×2
4
=
36 ×
= 1152
×
32
4
36
×32
72
1080
1152
2
⎛1 ⎞
2) Evaluate: ⎜⎜ ⎟⎟
⎜⎝ 4 ⎟⎠
×2
9
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
580
Worked Solution
What is Wrong Here?
Identify the Errors
3) Evaluate: –92
4) Evaluate: (–0.02)3
ADDITIONAL EXERCISES
1. Write in exponential notation:
a) 9 • 9 • 2 • 2 • 2
b) 10 × 6 × 6 × 6 × 6 × 3 × 3
2. Evaluate each of the following:
a) 73
b) (–6)2
⎛ 2⎞
c) ⎜⎜ ⎟⎟
⎜⎝ 5 ⎟⎠
3
d) (0.5)3
e) 82 + 8
⎛ 1⎞
f) ⎜⎜− ⎟⎟⎟
⎜⎝ 4 ⎠
2
g) 402
h) (2.1)2
i) (–9)3
j) (–1)15
3. Order the following from lowest to highest value: 23, –42, –23, (–4)2, –25
Correct Process