Download Factors and Prime Factorization

4.1
Factors and Prime Factorization
Goal: Write the prime factorization of a number.
Vocabulary
Prime number:
Composite
number:
Prime
factorization:
Factor tree:
Monomial:
Example 1
Writing Factors
A rectangle has an area of 18 square feet. Find all possible whole
number dimensions of the rectangle.
1. Write 18 as a product of two whole numbers in all possible ways.
p
18
The factors of 18 are
The area of a
rectangle can be
found using the
formula, Area length width.
width
length
62
p
18
p
18
.
2. Use the factors to find all rectangles with an area of 18 square
feet that have whole number dimensions. Then label the given
rectangles.
Chapter 4 • Pre-Algebra Notetaking Guide
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4.1
Factors and Prime Factorization
Goal: Write the prime factorization of a number.
Vocabulary
A whole number greater than 1 that has exactly
two whole number factors, 1 and itself
Prime number:
Composite
number:
A whole number greater than 1 that has more than
two whole number factors
Prime
factorization:
When you write a number as a product of prime
numbers, you are writing its prime factorization.
Factor tree:
A factor tree is a diagram used to write the prime
factorization of a number.
Monomial:
A monomial is a number, a variable, or the product
of a number and one or more variables raised to
whole number powers.
Example 1
Writing Factors
A rectangle has an area of 18 square feet. Find all possible whole
number dimensions of the rectangle.
1. Write 18 as a product of two whole numbers in all possible ways.
18 p 1 18
9 p 2 18
6 p 3 18
The factors of 18 are 1, 2, 3, 6, 9, and 18 .
The area of a
rectangle can be
found using the
formula, Area length width.
width
length
2. Use the factors to find all rectangles with an area of 18 square
feet that have whole number dimensions. Then label the given
rectangles.
2
3
9
6
1
18
62
Chapter 4 • Pre-Algebra Notetaking Guide
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Writing a Prime Factorization
Example 2
Write the prime factorization of 420.
One possible factor tree:
420
Write original number.
10 Write 420 as 10 p
p
Write 10 as
6
.
. Write
as
p6.
p
Write 6 as
.
Another possible factor tree:
420
Write original number.
6 Write 420 as 6 p
p
Write 6 as
10
.
. Write
p 10 .
p
Write 10 as
.
Both trees give the same result: 420 .
Answer: The prime factorization of 420 is
Example 3
as
.
Factoring a Monomial
Factor the monomial 24x 4 y.
24x 4 y Copyright © Holt McDougal. All rights reserved.
p x4y
Write 24 as
.
p y Write x 4 as
.
Chapter 4 • Pre-Algebra Notetaking Guide
63
Example 2
Writing a Prime Factorization
Write the prime factorization of 420.
One possible factor tree:
420
10 42
Write original number.
Write 420 as 10 p 42 .
Write 10 as 2 p 5 . Write 42 as
2 5 7 6
2 5 7 2 3
7 p6.
Write 6 as 2 p 3 .
Another possible factor tree:
420
Write original number.
6 70
Write 420 as 6 p 70 .
Write 6 as 2 p 3 . Write 70 as
2 3 7 10
2 3 7 2 5
7 p 10 .
Write 10 as 2 p 5 .
Both trees give the same result: 420 2 p 2 p 3 p 5 p 7 .
Answer: The prime factorization of 420 is 22 p 3 p 5 p 7 .
Example 3
Factoring a Monomial
Factor the monomial 24x 4 y.
24x 4 y 2 p 2 p 2 p 3 p x 4 y
Write 24 as 2 p 2 p 2 p 3 .
2 p 2 p 2 p 3 p x p x p x p x p y Write x 4 as x p x p x p x .
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Chapter 4 • Pre-Algebra Notetaking Guide
63
Checkpoint Write all factors of the number.
1. 28
2. 48
Tell whether the number is prime or composite. If it is composite,
write its prime factorization.
3. 97
4. 117
Factor the monomial.
5. 21n5
64
Chapter 4 • Pre-Algebra Notetaking Guide
6. 18x 2 y 3
Copyright © Holt McDougal. All rights reserved.
Checkpoint Write all factors of the number.
1. 28
2. 48
1, 2, 4, 7, 14, 28
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Tell whether the number is prime or composite. If it is composite,
write its prime factorization.
3. 97
4. 117
prime
composite; 32 p 13
Factor the monomial.
5. 21n5
3p7pnpnpnpnpn
64
Chapter 4 • Pre-Algebra Notetaking Guide
6. 18x 2 y 3
2p3p3pxpxpypypy
Copyright © Holt McDougal. All rights reserved.
4.2
Greatest Common Factor
Goal: Find the greatest common factor of two or more numbers.
Vocabulary
Common factor:
Greatest
common
factor (GCF):
Relatively
prime:
Example 1
Finding the Greatest Common Factor
Volunteers A high school asks for volunteers to help clean up
local highways on one Saturday each month. The group of
volunteers has 27 freshman, 18 sophomores, 36 juniors, and
45 seniors. What is the greatest number of groups that can be
formed if each group is to have the same number of each type of
student? How many freshman, sophomores, juniors, and seniors
will be in each group?
Solution
Method 1 List the factors of each number. Identify the greatest
number that is on every list.
Factors of 27:
Factors of 18:
Factors of 36:
Factors of 45:
Copyright © Holt McDougal. All rights reserved.
The common
factors are
.
The GCF is
.
Chapter 4 • Pre-Algebra Notetaking Guide
65
4.2
Greatest Common Factor
Goal: Find the greatest common factor of two or more numbers.
Vocabulary
Common factor: A common factor is a whole number that is a
factor of two or more nonzero whole numbers.
Greatest
common
factor (GCF):
The GCF is the greatest whole number that
is a factor of two or more nonzero whole numbers.
Relatively Two or more numbers are relatively prime if their
greatest common factor is 1.
prime:
Example 1
Finding the Greatest Common Factor
Volunteers A high school asks for volunteers to help clean up
local highways on one Saturday each month. The group of
volunteers has 27 freshman, 18 sophomores, 36 juniors, and
45 seniors. What is the greatest number of groups that can be
formed if each group is to have the same number of each type of
student? How many freshman, sophomores, juniors, and seniors
will be in each group?
Solution
Method 1 List the factors of each number. Identify the greatest
number that is on every list.
Copyright © Holt McDougal. All rights reserved.
Factors of 27:
1, 3, 9, 27
Factors of 18:
1, 2, 3, 6, 9, 18
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 45:
1, 3, 5, 9, 15, 45
The common
factors are
1, 3, and 9 .
The GCF is
9 .
Chapter 4 • Pre-Algebra Notetaking Guide
65
Method 2 Write the prime factorization of each number. The GCF
is the product of the prime factors.
27 18 The common prime
factors are
.
36 The GCF is
.
45 Answer: The greatest number of groups that can be formed is
Each group will have 27 sophomores, 36 freshman, 18 juniors, and 45 .
seniors.
Checkpoint Find the greatest common factor of the numbers.
1. 54, 81
Example 2
2. 12, 48, 66
Identifying Relatively Prime Numbers
Find the greatest common factor of the numbers. Then tell whether
the numbers are relatively prime.
a. 28, 63
b. 42, 55
Solution
a. List the factors of each number. Identify the greatest number
that the lists have in common.
Factors of 28:
Factors of 63:
The GCF is
. So, the numbers
relatively prime.
b. Write the prime factorization of each number.
42 The GCF is
66
Chapter 4 • Pre-Algebra Notetaking Guide
55 . So, the numbers
relatively prime.
Copyright © Holt McDougal. All rights reserved.
Method 2 Write the prime factorization of each number. The GCF
is the product of the prime factors.
27 3 p 3 p 3
18 2 p 3 p 3
The common prime
factors are 3 and 3 .
36 2 p 2 p 3 p 3
The GCF is 3 p 3 9 .
45 3 p 3 p 5
Answer: The greatest number of groups that can be formed is 9 .
Each group will have 27 9 3 freshman, 18 9 2
sophomores, 36 9 4 juniors, and 45 9 5 seniors.
Checkpoint Find the greatest common factor of the numbers.
1. 54, 81
2. 12, 48, 66
27
Example 2
6
Identifying Relatively Prime Numbers
Find the greatest common factor of the numbers. Then tell whether
the numbers are relatively prime.
a. 28, 63
b. 42, 55
Solution
a. List the factors of each number. Identify the greatest number
that the lists have in common.
Factors of 28:
1, 2, 4, 7, 14, 28
Factors of 63:
1, 3, 7, 9, 21, 63
The GCF is 7 . So, the numbers are not relatively prime.
b. Write the prime factorization of each number.
42 2 p 3 p 7
The GCF is 1 . So, the numbers
66
Chapter 4 • Pre-Algebra Notetaking Guide
55 5 p 11
are
relatively prime.
Copyright © Holt McDougal. All rights reserved.
Checkpoint Find the greatest common factor of the numbers.
Then tell whether the numbers are relatively prime.
3. 30, 49
Example 3
4. 52, 78
Finding the GCF of Monomials
Find the greatest common factor of 16x 2 y and 26x 2 y 3.
Solution
Factor the monomials. The GCF is the product of the common
factors.
16x 2 y 26x 2 y 3 Answer: The GCF is
.
Checkpoint Find the greatest common factor of the monomials.
5. 12x 3, 18x 2
Copyright © Holt McDougal. All rights reserved.
6. 40xy 3, 24xy
Chapter 4 • Pre-Algebra Notetaking Guide
67
Checkpoint Find the greatest common factor of the numbers.
Then tell whether the numbers are relatively prime.
3. 30, 49
4. 52, 78
1; relatively prime
Example 3
26; not relatively prime
Finding the GCF of Monomials
Find the greatest common factor of 16x 2 y and 26x 2 y 3.
Solution
Factor the monomials. The GCF is the product of the common
factors.
16x 2 y 2 p 2 p 2 p 2 p x p x p y
26x 2 y 3 2 p 13 p x p x p y p y p y
Answer: The GCF is 2x 2 y .
Checkpoint Find the greatest common factor of the monomials.
5. 12x 3, 18x 2
6. 40xy 3, 24xy
6x 2
Copyright © Holt McDougal. All rights reserved.
8xy
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67
4.3
Equivalent Fractions
Goal: Write equivalent fractions.
Vocabulary
Equivalent
fractions:
Simplest
form:
Equivalent Fractions
Words To write equivalent fractions, multiply or divide the
numerator and the denominator by the same nonzero number.
Algebra For all numbers a, b, and c, where b 0 and c 0,
a
apc
a
ac
and .
b
bpc
b
bc
1
3
1p2
3p2
2
22
1
6
62
3
2
6
Numbers Example 1
Writing Equivalent Fractions
6
18
Write two fractions that are equivalent to .
Multiply or divide the numerator and the denominator by the
.
6
6p2
18
18 p 2
Multiply numerator and denominator by 2.
6
63
18
18 3
Divide numerator and denominator by 3.
Answer: The fractions
68
Chapter 4 • Pre-Algebra Notetaking Guide
and
6
18
are equivalent to .
Copyright © Holt McDougal. All rights reserved.
4.3
Equivalent Fractions
Goal: Write equivalent fractions.
Vocabulary
Equivalent
fractions:
Two fractions that represent the same number are
called equivalent fractions.
Simplest A fraction is in simplest form when its numerator and
its denominator are relatively prime.
form:
Equivalent Fractions
Words To write equivalent fractions, multiply or divide the
numerator and the denominator by the same nonzero number.
Algebra For all numbers a, b, and c, where b 0 and c 0,
a
apc
a
ac
and .
b
bpc
b
bc
1
3
1p2
3p2
2
22
1
6
62
3
2
6
Numbers Example 1
Writing Equivalent Fractions
6
18
Write two fractions that are equivalent to .
Multiply or divide the numerator and the denominator by the
same nonzero number .
6
6p2
12
36
18
18 p 2
Multiply numerator and denominator by 2.
6
63
2
6
18
18 3
Divide numerator and denominator by 3.
12
2
6
Answer: The fractions 3
and 6 are equivalent to .
6
18
68
Chapter 4 • Pre-Algebra Notetaking Guide
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Checkpoint Write two fractions that are equivalent to the
given fraction.
7
14
10
25
4
16
1. 2. Example 2
3. Writing a Fraction in Simplest Form
8
36
Write in simplest form.
Write the prime factorizations of the numerator and denominator.
8
36 The GCF of 8 and 36 is
8
8 36
36 .
Divide numerator and denominator by GCF.
Simplify.
Checkpoint Write the fraction in simplest form.
3
18
4. Copyright © Holt McDougal. All rights reserved.
12
32
5. 24
42
6. Chapter 4 • Pre-Algebra Notetaking Guide
69
Checkpoint Write two fractions that are equivalent to the
given fraction.
7
14
10
25
4
16
1. 2. 1 14
, 2 28
Example 2
3. 1 8
, 4 32
2 20
, 5 50
Writing a Fraction in Simplest Form
8
36
Write in simplest form.
Write the prime factorizations of the numerator and denominator.
8 23
36 2 2 p 3 2
The GCF of 8 and 36 is 22 4 .
4
8
8 36
36 4
2
9
Divide numerator and denominator by GCF.
Simplify.
Checkpoint Write the fraction in simplest form.
12
32
3
18
4. 1
6
Copyright © Holt McDougal. All rights reserved.
24
42
5. 6. 3
8
4
7
Chapter 4 • Pre-Algebra Notetaking Guide
69
Example 3
Simplifying a Variable Expression
14 x2 y
Write in simplest form.
35x 3
14 x 2 y
35 x 3 Factor numerator and denominator.
Divide out common factors.
Simplify.
Checkpoint Write the variable expression in simplest form.
9a
15a
7. 2
70
Chapter 4 • Pre-Algebra Notetaking Guide
16mn2
28n
8. 39s t 2
3s t
9. 2
Copyright © Holt McDougal. All rights reserved.
Example 3
Simplifying a Variable Expression
14 x2 y
Write in simplest form.
35x 3
2p7pxpxpy
14 x 2 y
3 35 x
5p7pxpxpx
1
1
1
2p7
p
xp
xpy
5p7
p
xp
xpx
1
2y
5
x
1
Factor numerator and denominator.
Divide out common factors.
1
Simplify.
Checkpoint Write the variable expression in simplest form.
16mn2
28n
9a
15a
7. 2
8. 3
5a
70
Chapter 4 • Pre-Algebra Notetaking Guide
4mn
7
39s t 2
3s t
9. 2
13t
s
Copyright © Holt McDougal. All rights reserved.
4.4
Least Common Multiple
Goal: Find the least common multiple of two numbers.
Vocabulary
Multiple:
Common
multiple:
Least common
multiple (LCM):
Least common
denominator (LCD):
Example 1
Finding the Least Common Multiple
Find the least common multiple of 6 and 14.
Solution
You can use one of two methods to find the LCM.
Method 1 List the multiples of each number. Identify the least
number that is on both lists.
The LCM of
6 and 14 is
.
Multiples of 6:
Multiples of 14:
Method 2 Find the common factors of the numbers.
6
14 The common
factor is
.
Multiply all of the factors, using each common factor only once.
LCM Answer: Both methods get the same result. The LCM is
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.
Chapter 4 • Pre-Algebra Notetaking Guide
71
4.4
Least Common Multiple
Goal: Find the least common multiple of two numbers.
Vocabulary
Multiple:
A multiple of a whole number is the product of the
number and any nonzero whole number.
Common A multiple that is shared by two or more numbers
multiple: is a common multiple.
Least common The least common multiple of two or more
multiple (LCM): numbers
The least common multiple of the
Least common
denominator (LCD): denominators of two or more fractions
Example 1
Finding the Least Common Multiple
Find the least common multiple of 6 and 14.
Solution
You can use one of two methods to find the LCM.
Method 1 List the multiples of each number. Identify the least
number that is on both lists.
Multiples of 6:
Multiples of 14:
6, 12, 18, 24, 30, 36, 42, 48
14, 28, 42, 56
The LCM of
6 and 14 is
42 .
Method 2 Find the common factors of the numbers.
6 2p3
14 2 p 7
The common
factor is 2 .
Multiply all of the factors, using each common factor only once.
LCM 2 p 3 p 7 42
Answer: Both methods get the same result. The LCM is 42 .
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Chapter 4 • Pre-Algebra Notetaking Guide
71
Example 2
Finding the Least Common Multiple of Monomials
Find the least common multiple 6xy and 16x 2.
6xy 16x 2 LCM Answer: The least common multiple of 6xy and 16x 2 is
.
Checkpoint Find the least common multiple of the numbers or
the monomials.
1. 8, 18
2. 4, 5, 15
3. 12x, 18x 2
4. 4xy, 10xz 2
Example 3
Comparing Fractions Using the LCD
Summer Sports Last year, a summer resort had 165,000 visitors,
including 44,000 water skiers. This year, the resort had 180,000
visitors, including 63,000 water skiers. In which year was the
fraction of water skiers greater?
Solution
1. Write the fractions and simplify.
Number of water skiers
Total number of visitors
Number of water skiers
Total number of visitors
Last year: This year: 2. Find the LCD of
and
. The LCM of
. So, the LCD of the fractions is
72
Chapter 4 • Pre-Algebra Notetaking Guide
and
is
.
Copyright © Holt McDougal. All rights reserved.
Example 2
Finding the Least Common Multiple of Monomials
Find the least common multiple 6xy and 16x 2.
6xy 2 p 3 p x p y
16x 2 2 p 2 p 2 p 2 p x p x
LCM 2 p 2 p 2 p 2 p 3 p x p x p y 48x 2y
Answer: The least common multiple of 6xy and 16x 2 is 48x 2y .
Checkpoint Find the least common multiple of the numbers or
the monomials.
1. 8, 18
2. 4, 5, 15
72
60
3. 12x, 18x 2
4. 4xy, 10xz 2
36x 2
Example 3
20xyz 2
Comparing Fractions Using the LCD
Summer Sports Last year, a summer resort had 165,000 visitors,
including 44,000 water skiers. This year, the resort had 180,000
visitors, including 63,000 water skiers. In which year was the
fraction of water skiers greater?
Solution
1. Write the fractions and simplify.
Number of water skiers
44,000
4
Number of water skiers
63,000
7
Last year: 165,000
15
Total number of visitors
This year: 180,000
20
Total number of visitors
2. Find the LCD of 4 and 7 . The LCM of 15 and 20 is
15
20
60 . So, the LCD of the fractions is 60 .
72
Chapter 4 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
3. Write equivalent fractions using the LCD.
Last year:
This year:
<
4. Compare the numerators:
, so
<
.
Answer: The fraction of water skiers was greater
.
Ordering Fractions and Mixed Numbers
Example 4
5 9
12 2
33
8
Order the numbers 4 , , and from least to greatest.
1. Write the mixed number as an improper fraction.
5
12
4 12
12
9
2
33
8
2. Find the LCD of , , and . The LCM of 12, 2,
12
and 8 is
. So, the LCD is
.
3. Write equivalent fractions using the LCD.
p
9p
9
2p
12 p
12
2
33 p
33
8 p
8
<
4. Compare the numerators:
<
so
<
and
and
<
,
.
Answer: From least to greatest, the numbers are
,
Copyright © Holt McDougal. All rights reserved.
, and
.
Chapter 4 • Pre-Algebra Notetaking Guide
73
3. Write equivalent fractions using the LCD.
Last year:
16
4
4p4
60
15
15 p 4
This year:
21
7
7p3
60
20
20 p 3
4. Compare the numerators:
21
16
< , so
60
60
4
7
< .
15
20
Answer: The fraction of water skiers was greater this year .
Example 4
Ordering Fractions and Mixed Numbers
5 9
12 2
33
8
Order the numbers 4 , , and from least to greatest.
1. Write the mixed number as an improper fraction.
53
4 p 12 5
5
4 12
12
12
53 9
33
2. Find the LCD of , , and . The LCM of 12, 2,
8
12 2
and 8 is 24 . So, the LCD is 24 .
3. Write equivalent fractions using the LCD.
53 p 2
12
53
9p
106
108
9
24
24
2 p 12
12 p 2
12
2
3
33 p
33
8 p 3
8
99
24
4. Compare the numerators:
so
33
8
5
< 4 12
5
99
24
and 4 1
<
2
106
< 24
9
2
106
108
and 2
< ,
4
24
.
Answer: From least to greatest, the numbers are
33
5
, 4 , and
8
12
Copyright © Holt McDougal. All rights reserved.
9
2
.
Chapter 4 • Pre-Algebra Notetaking Guide
73
4.5
Rules of Exponents
Goal: Multiply and divide powers.
Product of Powers Property
Words To multiply powers with the same base, add their
exponents.
Algebra a m p a n a m n
Numbers 43 p 42 4
4
Using the Product of Powers Property
Example 1
a. 47 p 411 4
4
b. 2x 2 p 7x 6 2 p 7 p x 2 p x 6
Product of powers property
Add exponents.
Commutative property
of multiplication
2p7px
Product of powers property
2p7px
Add exponents.
Multiply.
Checkpoint Find the product. Write your answer using
exponents.
74
1. 25 p 212
2. (0.4)6 • (0.4)2 • (0.4)3
3. x 6 p x 13
4. b2 p b4 p b
Chapter 4 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
4.5
Rules of Exponents
Goal: Multiply and divide powers.
Product of Powers Property
Words To multiply powers with the same base, add their
exponents.
Algebra a m p a n a m n
Numbers 43 p 42 4
32
4
5
Using the Product of Powers Property
Example 1
a. 47 p 411 4
4
7 11
Product of powers property
18
b. 2x 2 p 7x 6 2 p 7 p x 2 p x 6
2p7px
26
2p7px
8
14x 8
Add exponents.
Commutative property
of multiplication
Product of powers property
Add exponents.
Multiply.
Checkpoint Find the product. Write your answer using
exponents.
1. 25 p 212
2. (0.4)6 • (0.4)2 • (0.4)3
217
3. x 6 p x 13
4. b2 p b4 p b
x 19
74
Chapter 4 • Pre-Algebra Notetaking Guide
(0.4)11
b7
Copyright © Holt McDougal. All rights reserved.
Quotient of Powers Property
Words To divide powers with the same base, subtract the
exponent of the denominator from the exponent of the
numerator.
am
a
mn
, where a 0
Algebra n a
57
5
Numbers 4 5
Using the Quotient of Powers Property
Example 2
(0.6)8
a. (0.6
(0.6)
)3
(0.6)
3x7
12x
5
Quotient of powers property
Subtract exponents.
3x
b. 3 12
3x
12
Quotient of powers property
Subtract exponents.
Divide numerator and denominator by
.
Checkpoint Find the quotient. Write your answer using
exponents.
59
5
(1.4)7
6. 4
4x 13
24 x
8. 11
5. 2
7. 9
Copyright © Holt McDougal. All rights reserved.
(1.4)
14 x 16
6x
Chapter 4 • Pre-Algebra Notetaking Guide
75
Quotient of Powers Property
Words To divide powers with the same base, subtract the
exponent of the denominator from the exponent of the
numerator.
am
a
mn
, where a 0
Algebra n a
57
5
Numbers 4 5
Example 2
74
5
3
Using the Quotient of Powers Property
(0.6)8
83
a. (0.6
Quotient of powers property
3 (0.6)
)
5
(0.6)
Subtract exponents.
73
3x7
3x
b. 3 12x
12
Quotient of powers property
4
3x
12
Subtract exponents.
4
x
4
Divide numerator and denominator by 3 .
Checkpoint Find the quotient. Write your answer using
exponents.
59
5
(1.4)7
6. 4
5. 2
(1.4)
57
14 x 16
6x
4x 13
24 x
7. 9
8. 11
x4
6
Copyright © Holt McDougal. All rights reserved.
(1.4)3
7x 5
3
Chapter 4 • Pre-Algebra Notetaking Guide
75
Using Both Properties of Powers
Example 3
4m3 p m4
12m
Simplify 2 .
4m3 p m4
4m
2
12m
12m2
4m
12m
2
4m
12
4m
12
Product of powers property
Add exponents.
Quotient of powers property
Subtract exponents.
Divide numerator and denominator
by
.
Checkpoint Simplify.
6m5 p m
15m
9. 3
76
Chapter 4 • Pre-Algebra Notetaking Guide
n2 p 10n6
5n
10. 3
Copyright © Holt McDougal. All rights reserved.
Example 3
Using Both Properties of Powers
4m3 p m4
12m
Simplify 2 .
34
4m3 p m4
4m
2
12m
12m2
Product of powers property
7
4m
12m
2
Add exponents.
72
4m
12
Quotient of powers property
5
4m
12
5
m
3
Subtract exponents.
Divide numerator and denominator
by 4 .
Checkpoint Simplify.
n2 p 10n6
5n
6m5 p m
15m
9. 3
10. 3
2m 3
5
76
Chapter 4 • Pre-Algebra Notetaking Guide
2n 5
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4.6
Negative and Zero Exponents
Goal: Work with negative and zero exponents.
Negative and Zero Exponents
For any nonzero number a, a0 1.
1
a
For any nonzero number a and any integer n, an n .
Powers with Negative and Zero Exponents
Example 1
Write the expression using only positive exponents.
a. 43 Definition of negative exponent
b. m5n0 m5 p
Definition of zero exponent
Definition of negative exponent
c. 13xy8 Definition of negative exponent
Checkpoint Write the expression using only positive exponents.
1. 33,3330
Example 2
2. 73
3. 2z2
4. 3x 4 y
Rewriting Fractions
Write the expression without using a fraction bar.
a. 1
15
Definition of negative exponent
a3
c
Definition of negative exponent
b. 5 Copyright © Holt McDougal. All rights reserved.
Chapter 4 • Pre-Algebra Notetaking Guide
77
4.6
Negative and Zero Exponents
Goal: Work with negative and zero exponents.
Negative and Zero Exponents
For any nonzero number a, a0 1.
1
a
For any nonzero number a and any integer n, an n .
Powers with Negative and Zero Exponents
Example 1
Write the expression using only positive exponents.
1
a. 43 3
Definition of negative exponent
b. m5n0 m5 p 1
Definition of zero exponent
4
1
5
Definition of negative exponent
m
13x
c. 13xy8 8
Definition of negative exponent
y
Checkpoint Write the expression using only positive exponents.
2. 73
1. 33,3330
3. 2z2
1
1
3
7
Example 2
Rewriting Fractions
4. 3x 4 y
2
2
z
3y
4
x
Write the expression without using a fraction bar.
1
15
Definition of negative exponent
a3
c
Definition of negative exponent
a. 151
b. 5 a 3c5
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Chapter 4 • Pre-Algebra Notetaking Guide
77
Checkpoint Write the expression without using a fraction bar.
1
18
1
100
5. Example 3
x5
y
3
c
6. 7. 2
8. 7
Using Powers Properties with Negative Exponents
Find the product or quotient. Write your answer using only
positive exponents.
0.7n4
b. a. 612 p 64
n
Solution
a. 612 p 64 6
6
0.7n4
b. 0.7n
n
Product of powers property
Add exponents.
Quotient of powers property
0.7n
Subtract exponents.
Definition of negative exponent
Checkpoint Find the product or quotient. Write your answer
using only positive exponents.
9. (0.3)10 p (0.3)7
78
Chapter 4 • Pre-Algebra Notetaking Guide
7d 4
d
10. 2
Copyright © Holt McDougal. All rights reserved.
Checkpoint Write the expression without using a fraction bar.
1
18
1
100
5. 181
Example 3
x5
y
3
c
6. 7. 2
1001 or 102
8. 7
3c2
x 5 y 7
Using Powers Properties with Negative Exponents
Find the product or quotient. Write your answer using only
positive exponents.
0.7n4
b. a. 612 p 64
n
Solution
a. 612 p 64 6
6
12 (4)
8
0.7n4
b. 0.7n
n
0.7n
5
0.7
n
5
Product of powers property
Add exponents.
4 1
Quotient of powers property
Subtract exponents.
Definition of negative exponent
Checkpoint Find the product or quotient. Write your answer
using only positive exponents.
9. (0.3)10 p (0.3)7
(0.3)3
78
Chapter 4 • Pre-Algebra Notetaking Guide
7d 4
d
10. 2
7
6
d
Copyright © Holt McDougal. All rights reserved.
4.7
Scientific Notation
Goal: Write numbers using scientific notation.
Using Scientific Notation
A number is written in scientific notation if it has the form c 10n
where 1 ≤ c < 10 and n is an integer.
Standard form
Product form
Scientific notation
725,000
7.25 100,000
7.25 10 5
0.006
6 0.001
6 103
Example 1
Writing Numbers in Scientific Notation
a. The average distance Mars is from the sun is 141,600,000
miles. Write this number in scientific notation.
Standard form
Product form
Scientific notation
b. The diameter of a quarter-ounce gold American Eagle coin is
0.022 meter. Write this number in scientific notation.
Standard form
Product form
Scientific notation
Example 2
Writing Numbers in Standard Form
a. Write 4.1 104 in standard form.
Scientific notation
Product form
Standard form
b. Write 7.23 106 in standard form.
Scientific notation
Product form
Standard form
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Chapter 4 • Pre-Algebra Notetaking Guide
79
4.7
Scientific Notation
Goal: Write numbers using scientific notation.
Using Scientific Notation
A number is written in scientific notation if it has the form c 10n
where 1 ≤ c < 10 and n is an integer.
Standard form
Product form
Scientific notation
725,000
7.25 100,000
7.25 10 5
0.006
6 0.001
6 103
Example 1
Writing Numbers in Scientific Notation
a. The average distance Mars is from the sun is 141,600,000
miles. Write this number in scientific notation.
Standard form
Product form
Scientific notation
141,600,000
1.416 100,000,000
1.416 108
b. The diameter of a quarter-ounce gold American Eagle coin is
0.022 meter. Write this number in scientific notation.
Standard form
Product form
Scientific notation
Example 2
2.2 102
2.2 0.01
0.022
Writing Numbers in Standard Form
a. Write 4.1 104 in standard form.
Scientific notation
Product form
4.1 104
4.1 10,000
b. Write 7.23 106 in standard form.
Scientific notation
Product form
7.23 10 6
Copyright © Holt McDougal. All rights reserved.
7.23 0.000001
Standard form
41,000
Standard form
0.00000723
Chapter 4 • Pre-Algebra Notetaking Guide
79
Checkpoint Write the number in scientific notation.
1. 3,050,000,000
2. 0.000082
Write the number in standard form.
4. 9.2 104
3. 6.53 107
Example 3
Ordering Numbers Using Scientific Notation
Order 5.3 105, 520,000, and 7.5 104 from least to greatest.
1. Write each number in scientific notation if necessary.
520,000 2. Order the numbers with different powers of 10.
< 10
Because 10
<
,
<
and
.
3. Order the numbers with the same power of 10.
Because
<
,
<
.
4. Write the original numbers in order from least to greatest.
;
;
Checkpoint Order the numbers from least to greatest.
5. 23,000; 3.4 103; 2.2 104
6. 4.5 104; 0.000047; 4.8 105
80
Chapter 4 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Checkpoint Write the number in scientific notation.
1. 3,050,000,000
2. 0.000082
8.2 105
3.05 10 9
Write the number in standard form.
4. 9.2 104
3. 6.53 107
65,300,000
Example 3
0.00092
Ordering Numbers Using Scientific Notation
Order 5.3 105, 520,000, and 7.5 104 from least to greatest.
1. Write each number in scientific notation if necessary.
520,000 5.2 10 5
2. Order the numbers with different powers of 10.
Because 10
4
< 10
5
, 7.5 10 4 < 5.3 10 5 and
7.5 10 4 < 5.2 10 5 .
3. Order the numbers with the same power of 10.
Because 5.2 < 5.3 , 5.2 10 5 < 5.3 10 5 .
4. Write the original numbers in order from least to greatest.
7.5 10 4 ; 520,000 ; 5.3 10 5
Checkpoint Order the numbers from least to greatest.
5. 23,000; 3.4 103; 2.2 104
3.4 103; 2.2 10 4; 23,000
6. 4.5 104; 0.000047; 4.8 105
0.000047; 4.8 105; 4.5 104
80
Chapter 4 • Pre-Algebra Notetaking Guide
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Multiplying Numbers in Scientific Notation
Example 4
Oxygen Atoms The volume of one mole of oxygen atoms is about
1.736 105 cubic meters. Find the volume of 1.5 104 moles of
oxygen atoms.
Solution
Number of
Volume of one mole
Total
moles
of oxygen atoms
volume
(
)(
(
)
)(
(
(
Substitute values.
)
)
)
Commutative and
associative properties
of multiplication
Multiply
and
.
Product of powers
property
Add exponents.
Answer: The volume of 1.5 104 moles of oxygen atoms is about
cubic meters.
Checkpoint Find the product. Write your answer in scientific
notation.
7. (2.5 103)(2 105)
Copyright © Holt McDougal. All rights reserved.
8. (1.5 102)(4 104)
Chapter 4 • Pre-Algebra Notetaking Guide
81
Multiplying Numbers in Scientific Notation
Example 4
Oxygen Atoms The volume of one mole of oxygen atoms is about
1.736 105 cubic meters. Find the volume of 1.5 104 moles of
oxygen atoms.
Solution
Number of
Volume of one mole
Total
moles
of oxygen atoms
volume
( 1.736 105 )( 1.5 104 )
( 1.736 1.5 )( 10
5
10
2.604 ( 105 104 )
2.604 ( 105 4 )
2.604 10 1
4
Substitute values.
)
Commutative and
associative properties
of multiplication
Multiply 1.736
and 1.5 .
Product of powers
property
Add exponents.
Answer: The volume of 1.5 104 moles of oxygen atoms is about
2.604 10 1 cubic meters.
Checkpoint Find the product. Write your answer in scientific
notation.
7. (2.5 103)(2 105)
5 108
Copyright © Holt McDougal. All rights reserved.
8. (1.5 102)(4 104)
6 106
Chapter 4 • Pre-Algebra Notetaking Guide
81
4
Words to Review
Give an example of the vocabulary word.
82
Prime number
Composite number
Prime factorization
Factor tree
Monomial
Common factor
Greatest common factor (GCF)
Relatively prime
Chapter 4 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
4
Words to Review
Give an example of the vocabulary word.
Prime number
3
Prime factorization
62p3
Composite number
8
Factor tree
420
10 42
2 5 7 6
2 5 7 2 3
Monomial
2x
Greatest common factor (GCF)
The GCF of 12 and 18 is 6.
82
Chapter 4 • Pre-Algebra Notetaking Guide
Common factor
2 is a common factor of
4 and 6.
Relatively prime
10 and 11 are relatively prime.
Copyright © Holt McDougal. All rights reserved.
Equivalent fractions
Simplest form
Multiple
Common multiple
Least common multiple
(LCM)
Least common denominator
(LCD)
Scientific notation
Review your notes and Chapter 4 by using the Chapter Review on
pages 212–215 of your textbook.
Copyright © Holt McDougal. All rights reserved.
Chapter 4 • Pre-Algebra Notetaking Guide
83
Equivalent fractions
12
36
Simplest form
2
6
The fractions and 6
18
are equivalent to .
Multiple
12 is a multiple of 6.
Least common multiple
(LCM)
The LCM of 12 and 18 is 36.
12
written in simplest form
36
1
is .
3
Common multiple
10 is a common multiple of
5 and 2.
Least common denominator
(LCD)
5
12
7
18
The LCD of and is 36.
Scientific notation
1.2 103
Review your notes and Chapter 4 by using the Chapter Review on
pages 212–215 of your textbook.
Copyright © Holt McDougal. All rights reserved.
Chapter 4 • Pre-Algebra Notetaking Guide
83