Download Chapter5 Sections 1 to 3

8/8/2012
Objectives
Identify bases and exponents
Multiply exponential expressions that have
like bases
Divide exponential expressions that have like
bases
Raise exponential expressions to a power
Find powers of products and quotients

Section 5.1

Rules for Exponents



Objective 1: Identify Bases and Exponents

Recall that an exponent indicates
repeated multiplication.

It indicates how many times the base is
used as a factor.
 For example, 35 represents the product of
five 3’s.

Objective 1: Identify Bases and Exponents

Expressions of the form xn are called exponential
expressions.

The base of an exponential expression can be a number, a
variable, or a combination of numbers and variables:
The base is 10. The exponent is 5. Read as “10 to the fifth
power” or simply as “10 to the fifth.”
In general, we have the following definition.

Natural-Number Exponents: A natural-number exponent tells
how many times its base is to be used as a factor.
For any number x and any natural number n,

When an exponent is 1, it is usually not written.
For example, 4 = 41 and x = x1.
1
8/8/2012
EXAMPLE 1

Identify the base and the exponent in each
expression:
a. 95, b. 7a3, c. (7a)3, d. –t10
Strategy To identify the base and exponent, we will look
for the form xy.
Why The exponent is the small raised number to the right
of the base.
Objective 2: Multiply Exponential
Expressions That Have Like Bases

To develop a rule for multiplying exponential expressions
that have the same base, we consider the product 62 × 63.


Since 62 means that 6 is to be
used as a factor two times,
and 63 means that 6 is to be
used as a factor three times,
we have:
We can quickly find this result if we keep the common base of 6
and add the exponents on 62 and 63.
EXAMPLE 1
Solution
a. In 95, the base is 9 and the exponent is 5.
b. 7a3 means 7a3. Thus, the base is a, not
7a. The exponent is 3.
c. Because of the parentheses in (7a)3, the
base is 7a and the exponent is 3.
d. Since the - symbol is not written within
parentheses, the base in -t10 is t and the
exponent is 10.
Objective 2: Multiply Exponential
Expressions That Have Like Bases

This example illustrates the following rule for
exponents.
 Product
Rule for Exponents: To multiply
exponential expressions that have the same
base, keep the common base and add the
exponents.
For any number x and any natural numbers m
and n, xm × xn = xm + n.
(Read as, “x to the mth power times x to the nth power
equals x to the m plus nth power.”)
2
8/8/2012
EXAMPLE 4

Geometry
Find an expression that
represents the area of the
rectangle.
EXAMPLE 4
Geometry
Solution
Strategy We will multiply the length of the rectangle
by its width.
The area of the rectangle is x8 square feet, which can be
written as x8 ft2.
Why The area of a rectangle is equal to the product
of its length and width.
Objective 3: Divide Exponential
Expressions That Have Like Bases

To develop a rule for dividing exponential expressions that
have the same base, we consider the quotient 45/42,
where the exponent in the numerator is greater than the
exponent in the denominator.

We can simplify this fraction by removing the common factors of 4
in the numerator and denominator:

We can quickly find this result if we keep the common base, 4, and
subtract the exponents on 45 and 42.
Objective 3: Divide Exponential
Expressions That Have Like Bases

This example suggests another rule for
exponents.
 Quotient
Rule for Exponents: To divide
exponential expressions that have the same
base, keep the common base and subtract the
exponents.
For any nonzero number x and any natural
numbers m and n, where m > n, xm/xn = xm − n.
In other words, “x to the mth power divided by x to the
nth power equals x to the m minus nth power.”
3
8/8/2012
Objective 3: Divide Exponential
Expressions That Have Like Bases

Recall that like terms are terms with exactly the
same variables raised to exactly the same
powers.
 To
add or subtract exponential expressions, they
must be like terms.
 To multiply or divide exponential expressions, only
the bases need to be the same.
EXAMPLE 5

Simplify each expression:
a. 2016/209, b. x9/x3, c. (7.5n)12/(7.5n)11, d. a3b8/ab5
Strategy In each case, we want to write an equivalent
expression using each base only once. We will use
the quotient rule for exponents to do this.
Why The quotient rule for exponents is used to divide
exponential expressions that have the same base.
EXAMPLE 5
EXAMPLE 5
Solution
Solution
Read as “20 to the sixteenth power divided by 20 to the ninth power.”
4
8/8/2012
Objective 4: Raise Exponential
Expressions to a Power
To develop another rule for exponents, we consider (53)4.

Here, an exponential expression, 53, is raised to a power.
 Since 53 is the base and 4 is the exponent, (53)4 can be written as
53 × 53 × 53 × 53.
 Because each of the four factors of 53 contains three factors of 5,
there are 4 × 3 or 12 factors of 5.


We can quickly find this result if we keep
the common base of 5 and multiply the
exponents.
EXAMPLE 7

Simplify:
a. (23)7, b. [(−6)2]5, c. (z8)8
Objective 4: Raise Exponential
Expressions to a Power

This example suggests the following rule for
exponents.
 Power
Rule for Exponents: To raise an
exponential expression to a power, keep the
base and multiply the exponents.
For any number x and any natural numbers m
and n, (xm)n = xm × n = xmn.
Read as, “the quantity of x to the mth power raised to
the nth power equals x to the mnth power.”
EXAMPLE 7
Solution
Read as “2 cubed raised to the seventh power.”
Read as “negative six squared raised to the fifth power.”
Strategy In each case, we want to write an equivalent
expression using one base and one exponent. We
will use the power rule for exponents to do this.
Why Each expression is a power of a power.
5
8/8/2012
Objective 5: Find Powers of Products and
Quotients
To develop more rules for exponents, we consider:

expression (2x)3, which is a power of the
product of 2 and x
 and the expression (2/x)3, which is a power of the
quotient of 2 and x.
 the
EXAMPLE 9

Simplify:
a. (3c)4, b. (x2y3)5, c. (−¼a3b)2
Objective 5: Find Powers of Products and
Quotients

These examples illustrate the following rules for
exponents.
 Powers
of a Product and a Quotient:
To raise a product to a power, raise each factor of the
product to that power.
To raise a quotient to a power, raise the numerator
and the denominator to that power.
For any numbers x and y, and any natural number n:
(xy)n = xnyn and (x/y)n = xn/yn, where y ≠ 0.
EXAMPLE 9
Solution
Strategy In each case, we want to write the expression
in an equivalent form in which each base is raised to
a single power. We will use the power of a product
rule for exponents to do this.
Why Within each set of parentheses is a product, and
each of those products is raised to a power.
6
8/8/2012
Objective 5: Find Powers of Products and
Quotients

The rules for natural-number exponents are
summarized as follows.
Section 5.2
 If
m and n represent natural numbers and there are no
divisions by zero, then
Objectives
Use the zero exponent rule
 Use the negative integer exponent rule
 Use exponent rules to change negative
exponents in fractions to positive
exponents
 Use all exponent rules to simplify
expressions

Zero and Negative
Exponents
Objective 1: Use the Zero Exponent Rule

We now extend the discussion of naturalnumber exponents to include exponents
that are zero and exponents that are
negative integers.
 To
develop the definition of a zero exponent,
we will simplify the expression 53/53 in two
ways and compare the results.
7
8/8/2012
EXAMPLE 1
Objective 1: Use the Zero Exponent Rule



First, we apply the quotient rule for exponents, where we
subtract the equal exponents in the numerator and
denominator. The result is 50.
In the second approach,we write 53 as 5 × 5 × 5 and remove
the common factors of 5 in the numerator and denominator.
The result is 1.

Strategy We note that each exponent is 0. To simplify the
expressions, we will identify the base and use the zeroexponent rule.
Why If an expression contains a
nonzero base raised to the 0
power, we can replace it with 1.
Since 53/53 = 50 and 53/53 = 1, we conclude that 50 = 1. This
observation suggests the following definition:

Zero Exponents: Any nonzero base raised to the 0 power is 1.
For any nonzero real number x, x0 = 1. Read as “x to the zero power equals 1.”
EXAMPLE 1
Solution
Simplify. Assume a ≠ 0:
a. (−8)0, b. (14/15)0, c. (3a)0, d. 3a0
Objective 2: Use the Negative Integer
Exponent Rule

To develop the definition of a negative exponent, we will simplify 62/65
in two ways and compare the results.

If we apply the quotient rule for exponents, where we subtract the greater
exponent in the denominator from the lesser exponent in the numerator, we
get 6−3.
 In the second approach, we remove the two common factors of 6 to get
1/63.

Since 62/65 = 6−3 and 62/65 = 1/63, we conclude that 6−3 = 1/63.
Note that 6−3 is equal to the reciprocal of 6 3.
This observation suggests the following definition:
Negative Exponents: For any nonzero real number x and any integer n,
x−n = 1/xn. In words, x−n is the reciprocal of xn.
8
8/8/2012
EXAMPLE 2

Express using positive exponents
and simplify, if possible:
a. 3−2, b. y−1, c. (−2)−3,
d. 5−2 − 10−2
EXAMPLE 2
Solution
Strategy Since each exponent is a negative number, we
will use the negative exponent rule.
Why This rule enables us to write an exponential
expression that has a negative exponent in an
equivalent form using a positive exponent.
Objective 3: Use Exponent Rules to Change Negative
Exponents in Fractions to Positive Exponents


Negative exponents can appear in the numerator and/or the
denominator of a fraction. To develop rules for such situations, we
consider the following example.
Objective 3: Use Exponent Rules to Change Negative
Exponents in Fractions to Positive Exponents

This example suggests the following rules.
 Changing
from Negative to Positive Exponents:
A factor can be moved from the denominator to the
numerator or from the numerator to the
denominator of a fraction if the sign of its exponent
is changed.
For any nonzero real numbers x and y, and any
integers m and n, 1/x−n = xn, and x−m/y−n = yn/xm.
We can obtain this result in a simpler way. In a−4/b−3, we can move
a−4 from the numerator to the denominator and change the sign of
the exponent, and we can move b−3 from the denominator to the
numerator and change the sign of the exponent.

These rules streamline the process when
simplifying fractions involving negative exponents.
9
8/8/2012
Objective 3: Use Exponent Rules to Change Negative
Exponents in Fractions to Positive Exponents


When a fraction is raised to a negative power, we can use rules
for exponents to change the sign of the exponent. For example,
we see that:
This process can be streamlined using the following rule.
EXAMPLE 5

Simplify: (4/m)−2
Strategy We want to write this fraction that is raised
to a negative power in an equivalent form that
involves a positive power. We will use the negative
exponent and reciprocal rules to do this.

Negative Exponents and Reciprocals: A fraction raised to a
power is equal to the reciprocal of the fraction raised to the
opposite power.
 For any nonzero real numbers x and y, and any integer n,
(x/y)−n = (y/x)n.
EXAMPLE 5
Solution
Why It is usually easier to simplify exponential
expressions if the exponents are positive.
Objective 4: Use All Exponent Rules to
Simplify Expressions

The rules for exponents involving products,
powers, and quotients are also true for zero and
negative exponents.
10
8/8/2012
Objective 4: Use All Exponent Rules to
Simplify Expressions
The rules for exponents are used to simplify
expressions involving products, quotients,
and powers.
 In general, an expression involving
exponents is simplified when:

 Each
base occurs only once.
powers are raised to powers
 There are no parentheses.
 There are no negative or zero exponents.
 No
EXAMPLE 6

Simplify. Do not use negative exponents in the
answer.
a. x5 × x−3, b. x3/x7, c. (x3)−2, d. (2a3b−5)3, e. (3/b5)−4
Strategy In each case, we want to write an equivalent
expression using one base and one positive exponent.
We will use rules for exponents to do this.
Why These expressions are not in simplest form. In
parts a and b, the base x occurs more than once. In
parts c, d, and e, there is a negative exponent.
EXAMPLE 6
EXAMPLE 6
Solution
Solution
11
8/8/2012
Objectives
Section 5.3
Scientific Notation
Objective 1: Convert from Scientific to
Standard Notation

Scientists often deal with extremely large
and small numbers.
 For
example, the distance
from the Earth to the sun is
approximately 150,000,000
kilometers.
 The influenza virus, which
causes flu symptoms of cough,
sore throat, and headache,
has a diameter of 0.00000256
inch.
Convert from scientific to standard
notation
 Write numbers in scientific notation
 Perform computations with scientific
notation

Objective 1: Convert from Scientific to
Standard Notation

The numbers 150,000,000 and 0.00000256
are written in standard notation, which is
also called decimal notation.
 Because
they contain many zeros, they are
difficult to read and cumbersome to work with in
calculations.
In this section, we will discuss a more
convenient form in which we can write such
numbers.
12
8/8/2012
Objective 1: Convert from Scientific to
Standard Notation

Scientific notation provides a compact way of
writing very large or very small numbers.
Objective 1: Convert from Scientific to
Standard Notation

Two examples of numbers written in scientific
notation are shown below.
 Note

A positive number is written in scientific notation when it is
written in the form N × 10n, where 1 ≤ N < 10 and n is an integer.

To write numbers in scientific notation, you need to be familiar
with powers of 10, like those listed in the table below.
that each of them is the product of a decimal
number (between 1 and 10) and a power of 10.

Objective 1: Convert from Scientific to
Standard Notation

For example, to convert 3.67 × 102, we recall that
multiplying a decimal by 100 moves the decimal point 2
places to the right.

To convert 2.158 × 10−3 to standard notation, we recall that
dividing a decimal by 1,000 moves the decimal point 3
places to the left.
A number written in scientific notation can be
converted to standard notation by performing the
indicated multiplication.
Objective 1: Convert from Scientific to
Standard Notation

In 3.67 × 102 and 2.158 × 10−3, the exponent gives the number of
decimal places that the decimal point moves, and the sign of the
exponent indicates the direction in which it moves.
Applying this observation to several other examples, we have:
13
8/8/2012
Objective 1: Convert from Scientific to
Standard Notation

The following procedure summarizes our
observations.
from Scientific to Standard
Notation:
1. If the exponent is positive, move the decimal
point the same number of places to the right as
the exponent.
2. If the exponent is negative, move the
decimal point the same number of places to the
left as the absolute value of the exponent.
EXAMPLE 1

Convert to standard notation:
a. 3.467 × 105, b. 8.9 × 10−4
 Converting
EXAMPLE 1
Solution
a. Since the exponent in 105 is 5, the decimal point moves 5
places to the right.
Strategy In each case, we need to identify the exponent
on the power of 10 and consider its sign.
Why The exponent gives the number of decimal places
that we should move the decimal point. The sign of the
exponent indicates whether it should be moved to the
right or the left.
Objective 2: Write Numbers in Scientific
Notation

To write a number in scientific notation
(N × 10n) we first determine N and then n.
 Note:
Thus, 3.467 ×
105
= 346,700.
b. Since the exponent in 10−4 is −4, the decimal point moves 4
places to the left.
The results from the next example
illustrate the following forms to use when
converting numbers from standard to scientific
notation.
Thus, 8.9 × 10−4 = 0.00089.
14
8/8/2012
EXAMPLE 2

Write each number in scientific notation:
a. 150,000,000, b. 0.00000256, c. 432 × 105
Strategy We will write each number as the product of a
number between 1 and 10 and a power of 10.
EXAMPLE 2
Solution
a. We must write 150,000,000 (the distance in kilometers from the
Earth to the sun) as the product of a number between 1 and 10 and
a power of 10. We note that 1.5 lies between 1 and 10. To obtain
150,000,000, we must move the decimal point in 1.5 exactly 8
places to the right.
This will happen if we multiply 1.5 by 108. Therefore,
Why Numbers written in scientific notation have the form
N × 10n.
EXAMPLE 2
EXAMPLE 2
Solution
Solution
b. We must write 0.00000256 (the diameter in inches of a flu virus)
as the product of a number between 1 and 10 and a power of 10.
We note that 2.56 lies between 1 and 10. To obtain 0.00000256, the
decimal point in 2.56 must be moved 6 places to the left.
c. The number 432 × 105 is not written in scientific notation
because 432 is not a number between 1 and 10. To write this
number in scientific notation, we proceed as follows:
This will happen if we multiply 2.56 by 10-6.
Therefore,
Written in scientific notation, 432 × 105 is 4.32 × 107.
15
8/8/2012
Objective 3: Perform Calculations with
Scientific Notation

Another advantage of scientific notation
becomes apparent when we evaluate
products or quotients that involve very large
or small numbers.
we express those numbers in scientific
notation, we can use rules for exponents to
make the calculations easier.
EXAMPLE 3

Astronomy
Except for the sun, the nearest star visible to the
naked eye from most parts of the United States is
Sirius. Light from Sirius reaches Earth in about 70,000
hours. If light travels at approximately 670,000,000
mph, how far from Earth is Sirius?
 If
Strategy We can use the formula d = rt to find the distance
from Sirius to Earth.
Why We know the rate at which light travels and the
time it takes to travel from Sirius to the Earth.
EXAMPLE 3
Astronomy
EXAMPLE 3
Astronomy
Solution
Solution
The rate at which light travels is 670,000,000 mph and the time
it takes the light to travel from Sirius to Earth is 70,000 hr. To
find the distance from Sirius to Earth, we proceed as follows:
We note that 46.9 is not between 0 and 10, so 46.9 × 1012 is not
written in scientific notation. To answer in scientific notation, we
proceed as follows.
Conclusion: Sirius is approximately 4.69 × 1013 or
46,900,000,000,000 miles from Earth.
16