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Pre-Algebra Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 4-1 Factors and Monomials
Lesson 4-2 Powers and Exponents
Lesson 4-3 Prime Factorization
Lesson 4-4 Greatest Common Factor (GCF)
Lesson 4-5 Simplifying Algebraic Fractions
Lesson 4-6 Multiplying and Dividing Monomials
Lesson 4-7 Negative Exponents
Lesson 4-8 Scientific Notation
Example 1 Use Divisibility Rules
Example 2 Use Divisibility Rules to Solve a Problem
Example 3 Find Factors of a Number
Example 4 Identify Monomials
Determine whether 435 is divisible by 2, 3, 5, 6, or 10.
Number
Divisible?
Reason
2
no
3
yes
5
6
yes
The ones digit is 5.
no
10
no
435 is not divisible by 2, so it
cannot be divisible by 6.
The ones digit is not 0.
The ones digit is 5 and 5 is not
divisible by 2.
The sum of the digits is
or 12 and 12 is divisible by 3.
Answer: So, 435 is divisible by 3 and 5.
Determine whether 786 is divisible by 2, 3, 5, 6, or 10.
Answer: 786 is divisible by 2, 3, and 6.
Student Elections Sonya is running for student council
president. She wants to give out campaign flyers with a
pen to each student in the school. She can buy “Vote
for Sonya” pens in packages of 5, 6, or 10. If there are
306 students in the school and she wants no pens left
over, which size packages should she buy?
Size
Yes/No
Reason
5
6
no
The ones digit of 306 is not 0 or 5.
yes
306 is divisible by 2 and 3, so it is
also divisible by 6. Therefore,
there would be no pens left over.
The ones digit is not 0.
no
10
Answer: Sonya should buy pens in packages of 6.
Transportation A class of 72 students is taking a
field trip. The transportation department can provide
vans that seat 5, 6, or 10 students. If the teacher
wants all vans to be the same size and no empty
seats, what size vans should be used?
Answer: Vans that seat 6 should be used.
List all the factors of 64. Use the divisibility rules to
determine whether 64 is divisible by 2, 3, 5, and so
on. Then use division to find other factors of 64.
Number
64 Divisible by Number?
2
yes
yes
3
no
4
yes
5
no
6
no
7
no
yes
1
8
Factor Pairs
___
___
___
___
Answer: So, the factors of 64 are 1, 2, 4, 8, 16, 32,
and 64.
List all the factors of 96.
Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96
Determine whether
is a monomial.
Distributive Property
Simplify.
Answer: This expression is not a monomial because
in its simplest form, it involves two terms
that are added.
Determine whether
is a monomial.
Answer: This expression is a monomial because
it is the product of a rational number,
and a variable, x.
,
Determine whether each expression is a monomial.
a.
Answer: monomial
b.
Answer: not a monomial
Example 1 Write Expressions Using Exponents
Example 2 Use Exponents in Expanded Form
Example 3 Evaluate Expressions
Write
using exponents.
Answer: The base is 6. It is a factor 4 times, so the
exponent is 4.
Write p using exponents.
Answer: The base is p. It is a factor 1 time, so the
exponent is 1.
Write (–1)(–1)(–1) using exponents.
Answer: The base is – 1. It is a factor 3 times, so the
exponent is 3.
Write
using exponents.
Answer: The base is
exponent is 2.
. It is a factor 2 times, so the
Write each expression using exponents.
Answer: First group the factors with like bases. Then
write using exponents.
Write each expression using exponents.
d.
a.
Answer:
Answer:
b.
e.
Answer:
Answer:
c.
Answer:
Express 235,016 in expanded form.
Answer:
Step 1 Use place value to write the value of each digit
in the number.
Step 2 Write each place value as a power of 10
using exponents.
Express 24,706 in expanded form.
Answer:
Evaluate
.
4 is a factor two times.
Multiply.
Answer: 16
Evaluate
if
.
Replace r with –2.
–2 is a factor 3 times.
Multiply.
Subtract.
Answer:
Evaluate
if
and
.
Replace x with 2 and y with –2.
Simplify the expression
inside the parentheses.
Evaluate (0)2.
Simplify.
Answer: 0
Evaluate each expression.
a.
Answer: 81
b.
if
Answer: 84
c.
Answer: –24
Example 1 Identify Numbers as Prime or Composite
Example 2 Write Prime Factorization
Example 3 Factor Monomials
Determine whether 31 is prime or composite.
Find factors of 31 by listing the whole number pairs
whose product is 31.
The number 31 has only two factors.
Answer: Therefore, 31 is a prime number.
Determine whether 36 is prime or composite.
Find factors of 36 by listing the whole number pairs
whose product is 36.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Answer: Since the number has more than two factors,
it is composite.
Determine whether each number is prime or composite.
a. 49
Answer: composite
b. 29
Answer: prime
Write the prime factorization of 56.
^
\
^
\ \
^
The prime factorization is complete because 2 and 7
are prime numbers.
Answer: The prime factorization of 56 is
.
Write the prime factorization of 72.
Answer:
Factor
Answer:
.
Factor
Answer:
.
Factor each monomial.
a.
Answer:
b.
Answer:
Example 1 Find the GCF
Example 2 Find the GCF
Example 3 Use the GCF to Solve Problems
Example 4 Find the GCF of Monomials
Example 5 Factor Expressions
Find the GCF of 16 and 24.
Method 1 List the factors.
factors of 16: 1, 2, 4, 8, 16
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Answer: The greatest common factor of 16 and 24 is 8.
Find the GCF of 16 and 24.
Method 2 Use prime factorization.
The GCF is the product of the common prime factors.
Again, the GCF of 16 and 24 is 8.
Answer: 8
Find the GCF of 18 and 30.
Answer: 6
Find the GCF of 28 and 35.
First, factor each number completely.
The common prime factor is 7.
Answer: The GCF of 28 and 35 is 7.
Find the GCF of 12, 48, and 72.
First, factor each number completely.
The common prime factors are 2, 2, and 3.
Answer: The GCF of 12, 48, and 72 is
.
Find the GCF of each set of numbers.
a. 24, 32
Answer: 8
b. 30, 42, 60
Answer: 6
Cookie Sale Parents donated 150 chocolate chip
cookies and 120 molasses cookies for a school bake
sale.
If the cookies are arranged on plates, and each plate has
the same number of chocolate chip cookies and the
same number of molasses cookies, what is the largest
number of plates possible?
Find the GCF of 150 and 120.
The common prime factors are 2, 3, and 5.
The GCF of 150 and 120 is
or 30.
Answer: So, 30 plates are possible.
Cookie Sale Parents donated 150 chocolate chip
cookies and 120 molasses cookies for a school bake
sale.
How many chocolate chip and molasses cookies will be
on each plate?
Answer: So, each plate will have 5 chocolate chip
cookies and 4 molasses cookies.
Apples There are 96 red apples and 72 green apples
to be placed in baskets.
a. If the apples are arranged in baskets, and each
basket has the same number of red apples and the
same number of green apples, what is the largest
number of baskets possible?
Answer: 24 baskets
b. How many red apples and green apples will be in
each basket?
Answer: 4 red apples and 3 green apples
Find the GCF of
and
.
Completely factor each expression.
Circle the common factors.
Answer: The GCF of
is
Find the GCF of
Answer:
and
.
Factor
.
First, find the GCF of 3x and 12.
The GCF is 3.
Now, write each term as a product of the GCF and its
remaining factors.
Distributive Property
Answer:
Factor
Answer:
.
Example 1 Simplify Fractions
Example 2 Simplify Fractions
Example 3 Simplify Fractions in Measurement
Example 4 Simplify Algebraic Fractions
Example 5 Simplify Algebraic Fractions
Write
in simplest form.
Factor the numerator.
Factor the denominator.
The GCF of 16 and 24 is
or 8.
Divide the numerator and
denominator by the GCF.
Simplest form
Answer:
Write
Answer:
in simplest form.
Write
in simplest form.
1
1
1
1
1
1
1
1
Divide the numerator and
the denominator by the
GCF,
.
Simplify.
Answer:
Write
Answer:
in simplest form.
Measurement 250 pounds is what part of 1 ton?
There are 2000 pounds in 1 ton.
Write the fraction
1
1
in simplest form.
1 1
1
1
1
1
Divide the numerator and
the denominator by the
GCF,
.
Simplify.
Answer: So, 250 pounds is
of a ton.
80 feet is what part of 40 yards?
Answer:
Simplify
.
1
1
1
1
1
1
Divide the numerator
and the denominator
by the GCF,
.
Simplify.
Answer:
Simplify
Answer:
.
Multiple-Choice Test Item
Which fraction is
A
B
written in simplest form?
C
D
Read the Test Item
In simplest form means that the GCF of the numerator
and denominator is 1.
Solve the Test Item
1
1
1
1
Factor.
1 1
Answer: C
1
1
Multiple-Choice Test Item
Which fraction is
written in simplest form?
A
C
Answer: D
B
D
Example 1 Multiply Powers
Example 2 Multiply Monomials
Example 3 Divide Powers
Example 4 Divide Powers to Solve a Problem
Find
.
The common base is 3.
Add the exponents.
Check
Answer:
Find
Answer:
.
Find
.
The common base is y.
Add the exponents.
Answer:
Find (3p4)(–2p3).
(3 • –2)(p4 • p3) Use the Commutative and
(3p4)(–2p3)
Associative Properties.
(–6)(p4+3)
The common base is p.
–6p7
Add the exponents.
Answer: –6p7
Find each product.
a.
Answer:
b.
Answer:
Find
.
The common base is 8.
Subtract the exponents.
Answer:
Find
.
The common base is x.
Subtract the exponents.
Answer:
Find
Answer:
Folding Paper If you fold a sheet of paper in half, you
have a thickness of 2 sheets. Folding again, you have
a thickness of 4 sheets. Continue folding in half and
recording the thickness. How many times thicker is a
sheet that has been folded 4 times than a sheet that
has not been folded?
Write a division expression to compare the thickness.
Subtract the exponents.
Answer: So, the paper is 16 times thicker.
Racing Car A can run at a speed of
miles per
hour and car B runs at a speed of
miles per hour.
How many times faster is car A than car B?
Answer: Car A is 2 times faster than car B.
Example 1 Use Positive Exponents
Example 2 Use Negative Exponents
Example 3 Use Exponents to Solve a Problem
Example 4 Algebraic Expressions with Negative
Exponents
Write
using a positive exponent.
Definition of negative exponent
Answer:
Write
using a positive exponent.
Definition of negative exponent
Answer:
Write each expression using a positive exponent.
a.
Answer:
b.
Answer:
Write
as an expression using a negative exponent.
Find the prime factorization of 125.
Definition of exponents
Definition of negative exponent
Answer:
Write
Answer:
as an expression using a negative exponent.
Physics An atom is an incredibly small unit of matter.
The smallest atom has a diameter of approximately
of a nanometer, or 0.0000000001 meter. Write the
decimal as a fraction and as a power of 10.
Write the decimal as
a fraction.
Definition of negative
exponent
Answer:
Write 0.000001 as a fraction and as a power of 10.
Answer:
Evaluate
.
if
Replace r with –4.
Definition of negative exponent
Find
Answer:
.
Evaluate
Answer:
if
.
Example 1 Express Numbers in Standard Form
Example 2 Express Numbers in Scientific Notation
Example 3 Use Scientific Notation to Solve a Problem
Example 4 Compare Numbers in Scientific Notation
Express
in standard form.
Move the decimal point
4 places to the right.
Answer: 43,950
Express
in standard form.
Move the decimal point
6 places to the left.
Answer: 0.00000679
Express each number in standard form.
a.
Answer: 2,614,000
b.
Answer: 0.000803
Express 800,000 in scientific notation.
The decimal point
moves 5 places.
The exponent is positive.
Answer:
Express 1,320,000 in scientific notation.
The decimal point
moves 6 places.
The exponent is positive.
Answer:
Express 0.0119 in scientific notation.
The decimal point
moves 2 places.
The exponent is negative.
Answer:
Express each number in scientific notation.
a. 65,000
Answer:
b. 3,024,000
Answer:
c. 0.00042
Answer:
Space The table shows the planets and their
distances from the Sun. Estimate how many
times farther Pluto is from the Sun than Mercury
is from the Sun.
Planet
Distance from the Sun (km)
Mercury
5.80 x 107
Venus
1.03 x 108
Earth
1.55 x 108
Mars
2.28 x 108
Jupiter
7.78 x 108
Saturn
1.43 x 109
Uranus
2.87 x 109
Neptune
4.50 x 109
Pluto
5.90 x 109
Explore You know that the distance from the Sun to
Pluto is
km and the distance from the
Sun to Mercury is
km.
Plan
To find how many times farther Pluto is from the
Sun than Mercury is from the Sun, find the ratio
of Pluto’s distance to Mercury’s distance. Since
you are estimating, round the distance
to
and round the distance
to
.
Solve
Divide
Answer:
So, Pluto is about 1.0  102 or 100 times
farther from the Sun than Mercury.
Examine
Use estimation to check the reasonableness
of the results.
Space Use the table to estimate how many times
farther Pluto is from the Sun than Earth is from
the Sun.
Planet
Distance from the Sun (km)
Mercury
5.80 x 107
Venus
1.03 x 108
Earth
1.55 x 108
Mars
2.28 x 108
Jupiter
7.78 x 108
Saturn
1.43 x 109
Uranus
2.87 x 109
Neptune
4.50 x 109
Pluto
5.90 x 109
Answer: 30 times
farther
Space The diameters of Mercury, Saturn, and Pluto
are
kilometers,
kilometers, and
kilometers, respectively. List the planets in
order of increasing diameter.
First, order the numbers according to their exponents.
Then, order the numbers with the same exponent by
comparing the factors.
Mercury
and Pluto
Saturn
Step 1
Compare the factors:
Step 2
Pluto
Mercury
Answer: So, the order is Pluto, Mercury, and Saturn.
Order the numbers
,
and
in decreasing order.
Answer:
,
,
,
,
, and
.
Explore online information about the
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