Download Exponents

Section 1.2
Exponents
Pre-Activity
Preparation
A Chain Letter
1st round: 31
Chain letters are generated every
day. If you send a chain letter to
three friends and they each send it
on to three friends, who each send it
on to three friends, how many chain
letters are sent?
2nd round: 32
3rd round: 33
How many chain letters are there after ten rounds?
What if you originally send the letter to seven friends? How many rounds will it take to reach 117,649
people?
Compound interest: Suppose you have invested $1000 in an interest bond. If you know the interest
rate per conversion period is 0.02 and that the account has been compounded 8 times, you can
figure out how much is in the account by calculating $1000(1.02)8. The account has $1171.66 after
8 conversions.
Scientific Notation:
In one year, light travels approximately 5.9 × 1012 miles.
Geometry:
The mass of a hydrogen atom is 1.7 × 10–24.
Find the volume of a cube if
each side measures 5 cm:
The speed of light is 1.86 × 105 miles per second.
One gram is 1.01 × 10–6 tons.
(5 cm) = 125 cm
3
Each of these examples demonstrates the use and utility of exponents.
3
Learning Objectives
• Demonstrate how the rules for exponentiation work
• Expand an exponential expression
• Evaluate exponential expressions
• Simplify exponential expressions
27
Chapter 1 — Evaluating Expressions
28
Terminology
New Terms
Previously Used
to
Learn
Commutative Property
base
constant
expanded form
Distributive Property
exponent
expression
power
factor
simplify
variable
Building Mathematical Language
A base is a number that is being used as a factor.
base
An exponent is the number of times the base factor appears
in a product of base factors.
4
5 = 5:
5
: 5
: 5 = 125

4 factors
6
a = a:
a : a
: a
:a
:a
6 factors of a
Power is another word for exponent.
Powers of 1 and 0
5
4
exponent
Other notations:
5^4 and (5)^4
This symbol ^ is called a caret and indicates
that the number following it is an exponent.
This notation is used when showing a
superscript (smaller and raised up) number
as an exponent is not possible. It is also often
used on calculators to denote the exponent
will follow.
• A base to the power one is the same as the base alone; the base is to be used as a factor 1 time:
41 = 4
(–7)1 = –7
x1 = x
Conversely, no exponent means an exponent of 1 (not zero):
• Any non-zero number raised to the zero power is 1:
• Two special cases:
8 = 81
30 = 1
–75 = (–75)1
(-99)0 = 1
y = y1
z0 = 1
0x
is undefined and 00 is undefined.
x
0
Exponential Notation
An expression written in exponential notation uses exponents to count or show how many times a base
is used as a factor. Exponent notation is prevalent in algebra to such an extent that there are special rules
that govern its use and simplification procedures. Evaluating expressions with exponents, interpreting
formulas with exponents, and having an understanding of the magnitude of a number squared or cubed or
presented in scientific notation are necessary for progress through your required math and science courses
as well as good preparation for understanding complex consumer information such as amortization.
NOTE: The idea that exponents are “counts” of factors will be extended beyond whole numbers to
include negative numbers, fractions, and variables.
29
Section 1.2 — Exponents
Properties and Principles of Exponents
The exponent rules presented below are used to simplify expressions with exponents. The first two rules
in the chart are to be used as definitions. The Product, Power, and Quotient Rules apply to LIKE BASES
ONLY. By the numeric examples in Table 1, you can see how the rules were derived.
Table 1
Exponent Rule
Zero Exponent
Rule
Negative
Exponent Rule
Product Rule
Symbolic
Representation
Numeric Example
x0 = 1
x -m =
A zero exponent shows no factors of
the base. Do not confuse this with a
base raised to an exponent of 1 such
as 61 = 6.
60 = 1
1
xm
1
1
5 = 2 =
and
5
25
-2
1
= 52 = 25
-2
5
x m x n = x m+n
7 2 : 73 =
(7 : 7)(7 : 7 : 7) =
(x ) = xm n
(5 ) =
(5 )(5 )(5 )(5 )=
Power Rule
Quotient Rule
:
xm
= x m–n
xn
2
2
2
2
52$4 = 58
53 5 : 5 : 5
=
52
5:5
= 53-2 = 51 = 5
53 125
Validate: 2 =
=5
5
25
2
Product to Power
Rule
n
(ab )
= a n : bn
Any number raised to a negative
exponent is defined to be the inverse
of the number to that exponent.
The negative sign of the exponent
indicates the idea of inverse
location, not opposite value.
When multiplying, make sure the
bases are the same number (or
variable) before adding exponents.
Think: multiplication is linked to
addition.
7 2+3 = 75
2 4
m n
Key Observations
(3 : 5) =
= (3 : 5)(3 : 5)
= 32 : 52
Validate: 152 = 32 : 52
225 = 9 : 25
This rule is often called the
“power to power” rule. Multiply
the exponents; keep the base
unchanged.
When the bases are the same,
division is performed by subtracting
exponents.
Division is repeated subtractions:
make the mental connection by
linking division with subtraction.
Product to power rule is used when
different bases are raised to the
same exponent. It is not the same
as the distributive property—no
addition is present—but it does
distribute the exponent over
multiplication. Use the commutative
property to rearrange factors.
continued on next page
Chapter 1 — Evaluating Expressions
30
Exponent Rule
Symbolic
Representation
Numeric Example
Key Observations
2
Quotient to
Power Rule
 2   2  2 
  =   
 3   3  3 
22 4
= 2 =
3
9
This rule allows fractions to be
raised to powers. Notice it is not
like the quotient rule because
the bases in the numerator and
2
22 4 denominator are different numbers.
2
Validate:   = 2 =
3
9
3
n
an
a
=
 
bn
b
m
Combined Rules
Combined rules show a logical
extension of the quotient to power
rule with the product to power rule.
Situations where exponent rules
are combined occur frequently in
algebra.
2
 ax 
am xm
=
 
bm y m
 by 
 33 s  36 s 2
 4 = 2 8
5t
 5t 
Expanding the rule table to include algebraic examples underscores the importance of being able to apply
the rules to numeric or algebraic expressions. In Table 2 you can again see how the rules were derived.
Table 2
Exponent Rule
Symbolic
Representation
Numeric Example
Algebraic Example
Zero Exponent
Rule
x0 = 1
290 = 1
p0 = 1
Negative
Exponent Rule
Product Rule
Power Rule
x -m =
1
xm
x m x n = x m+n
(xm ) = xm n
n
:
7 −3 =
1
1
and −3 = 73
3
7
7
y −3 =
1
1
and −3 = y 3
3
y
y
31 : 34 =
(3)(3 : 3 : 3 : 3) =
x3 : x 4 =
xxx : xxxx =
31+ 4 = 35
x 3+ 4 = x 7
(7 ) =
(7 )(7 )(7 )=
(n ) =
(n )(n )(n )(n )(n )=
7 4:3 = 712
n3$5 = n15
4 3
4
4
4
3 5
3
3
3
3
3
31
Section 1.2 — Exponents
Symbolic
Representation
Exponent Rule
Quotient Rule
xm
m–n
n = x
x
Numeric Example
105
=
102
10 : 10 : 10 : 10 : 10
=
10 : 10
= 10
Product to Power
Rule
n
(ab )
5 -2
= 10
3
p4
=
p9
=
p p p p
p p p p ppppp
= p 4-9 = p -5 =
(3 : 2 )
4
= a n : bn
Algebraic Example
=
= 34 : 24
(5 : y )
3
1
p5
=
= 53 : y 3 = 53 y 3
Validate:
6 = 1296 and
34 : 24 = 81 × 16 = 1296
= 125 y 3
4
Quotient to Power
Rule
n
2
an
a
  = n
b
b
Combined Rules
52 25
5
  = 2 =
4
16
4
(3 : 2 )
(3 : 2 )
m
5 2
 ax 
am xm
=
 
bm y m
 by 
3 4
5
2
25 32
=
=
 
y5 y5
 y
2
=
 2x 
22 x 2 4 x 2
=
= 2
 
2 2
9y
 3y  3 y
32 : 210
=
34 : 212
1
1
=
2
2
3 : 2 12
An expression is simplified when there is no more than one of each different base raised to a single
positive exponent.
Not Simplified
Simplified
2
 
3
4
9
2
1
 
 y
1
y5
5
25
22
8
x −2
(3 x)(9 x)
(− z )
1
x2
27x2
z8
2 4
Chapter 1 — Evaluating Expressions
32
Scientific Notation
One important application of exponents is scientific notation. Scientists work with very large
numbers, like the distance from one star to another, and with very small numbers, like the
weight of a single atom. Working with numbers with many zeroes can be very cumbersome.
We can choose to write these numbers in a shortened way that is easier to read and use for
n
calculations. Numbers written in scientific notation look like: b × 10 , where:
b is a number between 1 and 10 (mathematically, 1 ≤ b < 10 )
n (the exponent) is an integer and shows the number of decimal places the decimal must
be moved to show the number in standard notation. (Note: when a number in standard
notation is less than one, its exponent is negative in scientific notation—an application of
the negative exponent rule.)
Standard Form
Scientific Notation
THINK
4.58 is “b” and it’s between 1 and 10. The initial
number is greater than 1 so the exponent is
positive. The decimal moves 5 places:
458000
458000
4.58 × 105
5 4 3 2 1
so n is 5
6.7 is “b” and it’s between 1 and 10. The
initial number is less than 1 so the exponent is
negative. The decimal moves 6 places:
0.0000067
0 0000067
6.7 × 10–6
1 2 3 4 5 6
so n is −6
1. Write the following numbers in Scientific Notation.
a) 0.00029
b) 0.00000315
____________
____________
c) 702000000
____________
2 Write the following in standard form.
a) 3.12 × 104
b) 2.1 × 10–8
____________
____________
Look it up!
Use any resource to look up the values of the following:
1. Length of an Angstrom in meters
2. Planck’s constant
____________
____________
3. The mass of an electron
____________
4. One electron volt in ergs
____________
c) 3.721 × 107
____________
33
Section 1.2 — Exponents
Models
Model 1
Note: To write an exponential expression in expanded form means to rewrite it without using
exponents.
Problem
1
23 : 32
Expanded Form
Evaluated
Validated
2:2:2:3:3
2
:2:2:3:3=
23 = 2 : 2 : 2 = 8
4
The bases are not the same
so evaluate each base with
its corresponding exponent
separately.
= 4
:2:3:3
and 32 = 9
= 8
:3:3
8 : 9 = 72
8
24
= 24 : 3
= 72
2
3
5 : 3–2
(22 )3
24
5 : 3−2
1
=5: 2
3
1
=5:
3:3
(22 )3
24
(22 )(22 )(22 )
=
24
2:2:2:2:2:2
=
2:2:2:2
5:
1
1
5
=5:
=
2
3
3:3 9
The bases are the same
number; use the power
rule and the quotient rule:
(22 )3
24
2 2 : 3 26
= 4 = 4
2
2
6− 4
=2
Go backwards:
5 5
= 2 = 5 : 3−2
9 3
(22 )3
24
(22 )(22 )(22 )
=
24
2:2: 2 : 2 : 2 : 2
=
2: 2: 2: 2
= 22 = 4
= 22 = 4
Model 2
Problem
1
(5x2)3
Expanded Form
Simplified
Validated
(5xx)3
(5x2)3
(5x2)3
=(5xx)(5xx)(5xx)
= 53x6
= (5x2) (5x2) (5x2)
= 125x6
= 5 : 5 : 5 : x2 : x2 : x2
= 125x6
Chapter 1 — Evaluating Expressions
34
Problem
2
Expanded Form
x 2 : x −5
x 2 : x −5
x 2 : x −5
x2
x5
= x 2+ ( −5)
=
=
3
23 x 2
2x
Simplified
x 2 : x −5
x2
= x 2−5 = x −3
5
x
1
= 3
x
=
= x −3
1
= 3
x
x:x
x:x:x:x:x
23 x 2 2 : 2 : 2 : x : x
=
2x
2: x
Validated
23 x 2
=
2x
 23  x 2 
=   
 2  x 
23 x 2
2x
2:2: 2 : x : x
=
1
2 : x
= 23−1 x 2−1
= 22 x = 4 x
= 22 x = 4 x
???
Why can we do this?
???
Why can we do this?
In many of the previous cases, there is more than one way of thinking about
the simplification process. You can use the rules for exponents or you can
apply rules you have already learned (or both). You can use the fact that the
exponent applies to the expression within parentheses or multiply within
parentheses first: the result is the same. Work the problem one way and then
use an alternative approach to validate your answer.
Addressing Common Errors
Issue
Not determining
the correct base
when simplifying
negative
exponent
expressions
Not remembering
to include the
constant factor
when raising an
expression to a
power
Incorrect
Process
3 x −5 =
1
3x5
(6x3)2 = 6x6
Resolution
Write implied
operations with
their appropriate
symbol to
emphasize each
base.
Correct Process
None required.
For 3x–5, the
implied operation
is multiplication.
3 : x −5 =
1
3
3: 5 = 5
x
x
1
−5
3 :x =
Apply each
(6x3)2
1 3
exponent rule
= (6x
)
= 313): (6x
5
step-by-step,
= 6 : 6 : xx3 : x3
showing the
36
appropriate factors = 36x
= 5
x
for constants as
well as variables.
Validation
(6x3)2
= (61 x3)2
= (61 : 2 x3 : 2)
= 62 x6
= 36x6 
35
Section 1.2 — Exponents
Issue
Incorrect
Process
Assuming that a
number raised
to a negative
exponent is
always a negative
number
10−3 = −1000 or
1
10−3 = −
1000
Resolution
Use the definition
of negative
exponents:
1
bm
1
The result
2−1 = is not
21
always− mnegative.
b = m
For example:
b
1
2−1 =
2
b−m =
Correct Process
Validation
10−3 =
1
1
=
3
10 1000
or
= 0.001
1
1
= 3 = 10-3 
1000 10
a positive number.
Combining
bases as well as
exponents when
multiplying with
unlike bases
3 7 = 21
5 4
9
The exponent
rules apply to LIKE
BASES ONLY.
The expression
3574 is already in
simplified form
because the bases
are distinct (3 and
7)
Evaluate it with a
calculator.
35 = 243
74 = 2401
243 : 2401 =583,443 
whereas
219 = 794,280,046,581
Misapplying the
power to power
rule
(53)2 = 55
Write out the
factors when
unsure of the
exponents rules.
(53)2 =
Evaluate with a
(53)(53) =
calculator:
(5 : 5 : 5)(5 : 5 : 5) (53)2 = 15,625
= 56
55 = 125
(125)2 = 15,625 
Preparation Inventory
Before proceeding, make sure that you can:
Expand exponential expressions
Evaluate numerical exponent expressions
Evaluate any number raised to the zero power
Understand how to apply a negative exponent to a number or variable
Use the exponent rules to simplify expressions in exponential form
Section 1.2
Activity
Exponents
Performance Criteria
• Writing exponential expressions in expanded form
– appropriate base factors
– correct number of factors of each base
– validation of the answer with alternative interpretation
• Evaluating numeric exponential expressions
– accuracy
– demonstrated use of exponent rules
– validation of the answer
• Simplifying algebraic expressions containing exponents
– demonstrated use of exponent rules
– answer presented in its simplest form
– validation of the answer
Critical Thinking Questions
1. How do you determine the base(s) in an exponential expression?
2. How do you use the exponent in expanding an exponential expression?
3. How do you validate that an expression is simplified?
36
Section 1.2 — Exponents
37
4. How does the name given to the Product to Power rule relate to the meaning of the rule?
5. By analyzing the common errors, what is the relationship between parentheses and the Exponent Rules
that use them?
6. How does the Quotient Rule for exponents justify the rule that any non-zero number raised to the zero
power is equal to 1?
7. Are there significant differences between the numeric examples and algebraic examples in Table 2? Explain
your answer.
Tips
for
Success
• Show your work one step at a time as you are developing the solution to a problem
• Start with the simplest possible examples and then move on to more complex ones
• When working a complex problem, model it using simpler or more concrete components. For example, if
x and y are used, see what happens if you exchange 2 for x and 3 for y.
Chapter 1 — Evaluating Expressions
38
Demonstrate Your Understanding
1. Use mental math to evaluate the following:
a) 31
b) 05
c) 17
d) (9 – 7)0
answer:
answer:
answer:
answer:
2. Use the definition of negative exponents to rewrite the following with positive exponents:
1
c)
a) 4–1
b) 3–2
d) 32 x −6
x −3
answer:
answer:
answer:
answer:
3. Evaluate the following exponential expressions.
Problem
a) 34 : 35
b) 3223
c)
d)
(2 )
6 4
(3 )
−3 2
35
 23 
e)  2 
3 
2
Expanded Form
Worked Solution
Validation
39
Section 1.2 — Exponents
4. Simplify the following algebraic expressions:
Problem
Expanded Form
Worked Solution
a) (x3)(x4)(x)
b)
(2x )
5 3
 3a 5 
c)  0 
 6b 
2
d) 5x –3
e) (5x)–3
f)
(3 b ): (3
4
−2
−5
b3 )
5. Extend the concept: What values for x make the statement true?
1
125
a) 2 x = 8
b) 5 x =
c) 3x : 32 = 1
d) 42 x = 46
Validation
Chapter 1 — Evaluating Expressions
40
Identify
and
Correct
the
Errors
Identify and correct the errors in the following problems.
Worked Solution
1) 7(10–3) =
=7(–10)(–10)(–10)
= –7000
2) (16x3)2 =
= 16x3+2
= 16x5
3) 43 52 =
= 205
2
42 16 4
4
=
4)   = 2 =
5
25 5
5
5)
24 x 7
= 6x2
5
4x
List the Errors
Correct Process
Validation