Transcript Chapter 4 - Crestwood Local Schools

Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Karen placed 56 bottles into boxes that each held 6 bottles. How
many boxes did she use?
10
4-1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
(For help, go to Skills Handbook p. 760.)
Find each quotient.
1. 480 ÷ 3
2. 365 ÷ 5
4. 288 ÷ 6
5.
3. 459 ÷ 9
354
2
6.
354
3
Check Skills You’ll Need
4-1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Solutions
1.
160
3 480
–3
18
–18
0
2.
73
5 365
–35
15
–15
0
4-1
3.
51
9 459
–45
9
– 9
0
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Solutions (continued)
4.
48
6 288
–24
48
–48
0
5.
177
2 354
–2
15
–14
14
–14
0
4-1
6.
118
3 354
–3
5
–3
24
–24
0
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Is the first number divisible by the second?
a. 1,028 by 2
Yes; 1,028 ends in 8.
b. 572 by 5
No; 572 doesn’t end in 0 or 5.
c. 275 by 10
No; 275 doesn’t end in 0.
Quick Check
4-1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Is the first number divisible by the second?
a. 1,028 by 3
No; 1 + 0 + 2 + 8 = 11; 11 is not divisible by 3.
b. 522 by 9
Yes; 5 + 2 + 2 = 9; 9 is divisible by 9.
Quick Check
4-1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Ms. Washington’s class is having a class photo
taken. Each row must have the same number of students.
There are 35 students in the class. How can Ms. Washington
arrange the students in rows if there must be at least 5
students, but no more than 10 students, in each row?
1 • 35, 5 • 7 Find pairs of factors of 35.
There can be 5 rows of 7 students, or 7 rows of 5 students.
Quick Check
4-1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
State whether each number is divisible by 2, 3, 5, 9, or 10.
1. 18
2, 3, 9
2. 90
2, 3, 5, 9, 10
5. List the positive factors of 36.
1, 2, 3, 4, 6, 9, 12, 18, 36
4-1
3. 81
3, 9
4. 25
5
Exponents
PRE-ALGEBRA LESSON 4-2
Which word best completes the statement: sometimes,
always, or never?
If the product of two factors is zero, both factors
are
zero.
sometimes
4-2
Exponents
PRE-ALGEBRA LESSON 4-2
(For help, go to Lesson 1-9.)
Find each product.
1. 3 • 3 • 3 • 3
2. –12 • (–12)
3. (–4)(–4)(–4)
4. 10 • 10 • 10 • 10
Check Skills You’ll Need
4-2
Exponents
PRE-ALGEBRA LESSON 4-2
Solutions
1. 3 • 3 • 3 • 3 = 81
2. –12 • (–12) = 144
3. (–4)(–4)(–4) = –64
4. 10 • 10 • 10 • 10 = 10,000
4-2
Exponents
PRE-ALGEBRA LESSON 4-2
Write using exponents.
a. (–11)(–11)(–11)(–11)
(–11)4
Include the negative sign
within parentheses.
b. –5 • x • x • y • y • x
–5 • x • x • x • y • y
Rewrite the expression using the
Commutative and Associative Properties.
–5x3y2
Write x • x • x and y • y using
exponents.
Quick Check
4-2
Exponents
PRE-ALGEBRA LESSON 4-2
Suppose a certain star is 104 light-years from Earth.
How many light-years is that?
104 = 10 • 10 • 10 • 10
= 10,000 light-years
The exponent indicates that the base 10 is
used as a factor 4 times.
Multiply.
Quick Check
4-2
Exponents
PRE-ALGEBRA LESSON 4-2
a. Simplify 3(1 + 4)3.
3(1 + 4)3 = 3(5)3
Work within parentheses first.
= 3 • 125
Simplify 53.
= 375
Multiply.
b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6.
7(w + 3)3 + z = 7(–5 + 3)3 + 6
Replace w with –5 and z with 6.
= 7(–2)3 + 6
Work within parentheses.
= 7(–8) + 6
Simplify (–2)3.
= –56 + 6
Multiply from left to right.
= –50
Add.
4-2
Quick Check
Exponents
PRE-ALGEBRA LESSON 4-2
Write using exponents.
1. x • y • z • x • z
x2yz2
2. a • b • b • b • 3
3ab3
3. Simplify 5(2 + 4)2.
180
4. Evaluate (g3 – 7)2 • 5 + 4, for g = 3.
2,004
4-2
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Find three integers whose sum is 12 and whose product is 42.
2, 3, 7
4-3
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
(For help, go to Lesson 4-1.)
List the positive factors of each number.
1. 15
2. 35
3. 7
4. 20
5. 100
6. 121
Check Skills You’ll Need
4-3
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Solutions
1. 1 • 15, 3 • 5;
1, 3, 5, 15
2. 1 • 35, 5 • 7;
1, 5, 7, 35
3. 1 • 7;
1, 7
4. 1 • 20, 2 • 10, 4 • 5;
1, 2, 4, 5, 10, 20
5. 1 • 100, 2 • 50, 4 • 25, 5 • 20, 10 • 10;
6. 1 • 121, 11 • 11;
1, 11, 121
4-3
1, 2, 4, 5, 10, 20, 25, 50, 100
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
State whether each number is prime or composite.
Explain.
a. 46
Composite; 46 has more than two factors, 1, 2, 23, and 46.
b. 13
Prime; 13 has exactly 2 factors, 1 and 13.
Quick Check
4-3
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Use a factor tree to write the prime factorization of
273.
273
Prime
Prime
3 • 7 • 13
3
•
91
7
•
Start with a prime factor.
Continue branching.
13
Stop when all factors are
prime.
Write the prime factorization.
273 = 3 • 7 • 13
Quick Check
4-3
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Find the GCF of each pair of numbers or expressions.
a. 24 and 30
24 = 23 • 3
Write the prime factorizations.
30 = 2 • 3 • 5
Find the common factors.
GCF = 2 • 3
Use the lesser power of the common
factors.
=6
The GCF of 24 and 30 is 6.
b. 36ab2 and 81b
36ab2 = 22 • 32 • a • b2
Write the prime factorizations.
81b =
34 • b
Find the common factors.
GCF = 32 • b
Use the lesser power of the
common factors.
= 9b
The GCF of 36ab2 and 81b is 9b.
4-3
Quick Check
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Tell whether each number is prime or composite.
1. 123
2. 47
composite
prime
Write the prime factorization for each number.
3. 64
4. 45
32 • 5
26
Find the GCF for each pair.
5. 80 and 120
40
6. 62b3c2d and 31b2c3d
31b2c2d
4-3
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Find the GCF for each pair of numbers.
a. 12 and 18
6
b. 15 and 60
c. 54 and 60
15
6
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
(For help, go to Lesson 4-3.)
Find each GCF.
1. 14, 21
2. 48, 60
3. 5mn, 15m2n
4. 63r2, 48s3
Check Skills You’ll Need
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Solutions
1. 14, 21
14 = 2 • 7
21 = 3 • 7
GCF = 7
2. 48, 60
48 = 24 • 3
60 = 22 • 3 • 5
GCF = 22 • 3 = 12
3. 5mn, 15m2n
5mn = 5 • m • n
15m2n = 3 • 5 • m2 • n
GCF = 5 • m • n = 5mn
4. 63r2, 48s3
63r2 = 32 • 7 • r2
48s3 = 24 • 3 • s3
GCF = 3
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Find two fractions equivalent to
a.
18
21
18
.
21
18 • 2
= 21 • 2
36
= 42
b.
18
21
=
18 ÷ 3
21 ÷ 3
= 6
7
The fractions
6
36
18
and
are both equivalent to
.
7
42
21
Quick Check
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
You learn that 21 out of the 28 students in a class,
or 21 , buy their lunches in the cafeteria. Write this fraction in
28
simplest form.
The GCF of 21 and 28 is 7.
21 = 21 ÷ 7
28
28 ÷ 7
= 3
4
Divide the numerator and denominator
by the GCF, 7.
Simplify.
3
of the students in the class buy their lunches in the cafeteria.
4
Quick Check
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Write in simplest form.
p
a. 2p
p
p1
= 1
2p
2p
=
1
2
Divide the numerator and denominator by
the common factor, p.
Simplify.
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
(continued)
b.
14q2rs3
8qrs2
14q2rs3
8qrs2
=
2•7•q•q•r•s•s•s
2•2•2•q•r•s•s
Write as a product of
prime factors.
=
21 • 7 • q1 • q • r 1 • s 1 • s 1 • s
21 • 2 • 2 • q1 • r1 • s1 • s1
Divide the numerator
and denominator by
the common factors.
=
7•q•s
2•2
Simplify.
=
7•q•s
4
Simplify.
=
7qs
4
Quick Check
4-4
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Find two fractions equivalent to each fraction.
1.
11
16
2.
7
21
Sample answer:
Sample answer:
22
33
and
32
48
1
10
and
3
30
Write in simplest form.
3. 13
52
1
4
4 8
4. wx2 y3
wxy
xy7
w
4-4
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Choose the symbol <, = ,or > that makes each statement true.
a. 2
5
8
+1+2
3
7
?4 +1 1
8
8
8
b. 3 5 + 2 1 + 9 ? 4 + 3 3
6
=
<
4-5
10
10
4
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(For help, go to Skills Handbook p. 775.)
Compare. Use > to < to complete each statement.
1. 3
0
2. –16
–25
3. 0
1
4. –30
–20
Check Skills You’ll Need
4-5
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(For help, go to Skills Handbook p.775.)
Solutions
1. 3 > 0
2. –16 > –25
3. 0 < 1
4. –30 < –20
4-5
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Aaron, Chris, Maria, Sonia, and Ling are on a class
committee. They want to choose two members to present their
conclusions to the class. How many different groups of two members
can they form?
4-5
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(continued)
First, pair Aaron with each of the four other committee members.
Next, pair Chris with each of the three members left.
Since Aaron and Chris have already been paired, you don’t need to
count them again. Repeat for the rest of the committee members.
Each successive tree has one less branch.
Aaron
Chris
Maria
Sonia
Ling
Chris
Maria
Sonia
Ling
Maria
Sonia
Ling
Sonia
Ling
There are 10 different groups of two committee members.
4-5
Quick Check
Problem Solving Strategy: Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Solve each problem.
1. Twelve people are at a party. Each person greets each of the other
persons exactly once. How many greetings will there be in all?
66
2. How many different pairs of classmates can you choose from six
classmates?
15 pairs
3. Each small box is a square. What is the number of different squares
shown?
17
4-5
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Write 29,716 in simplest form.
52,003
4
7
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
(For help, go to Lesson 4-4.)
Write in simplest form.
1. 2
10
3.
28
35
2. 14
21
4.
6
8
Check Skills You’ll Need
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Solutions
2
2÷2
1
=
=
10
10 ÷ 2
5
2.
14
14 ÷ 7
2
=
=
21
21 ÷ 7
3
3. 28 = 28 ÷ 7 = 4
35
35 ÷ 7
5
4.
6
6÷2
3
=
=
8
8÷2
4
1.
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Write two lists of fractions equivalent to 2.
3
2
= 4 = 6 = … Numerators and denominators are positive.
3
6
9
2
= –2 = –4 = … Numerators and denominators are negative.
3
–3
–6
Quick Check
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Graph each rational number on a number line.
3
a. – 4
b. 0.5
c. 0
d. 1
3
Quick Check
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
A fast sports car can accelerate from a stop to 90 ft/s
in 5 seconds. What is its acceleration in feet per second per
second (ft/s2)? Use the formula a = f – i , where a is
t
acceleration, f is final speed, i is initial speed, and t is time.
a=
=
f–i
t
Use the acceleration formula.
90 – 0
5
Substitute.
90
= 5
Subtract.
= 18
Write in simplest form.
The car’s acceleration is 18 ft/s2.
Quick Check
4-6
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Write three fractions equivalent to the given fraction.
–5
5
1. 6
10
15
Sample: –6 , 12 , 18
Graph each rational number on a number line.
2. a. 1
4
b. – 1
c. 1.5
2
d. 0.4
3. A car can accelerate from 0 to 70 ft/s in 5 s. What is the acceleration
of the car in feet per second per second (ft/s2)?
14 ft/s2
4-6
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
23  32 is the prime factorization for __.
?
72
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
(For help, go to Lesson 4-2.)
Write using exponents.
1. k • k • k • k
2. m • n • m • n
3. 2 • 2 • 2 • 2
4. 5 • 5 • 5
Check Skills You’ll Need
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Solutions
1. k • k • k • k = k4
2. m • n • m • n = m • m • n • n = m2n2
3. 2 • 2 • 2 • 2 = 24
4. 5 • 5 • 5 = 53
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression.
a. 52 • 53
52 • 53 = 52 + 3
Add the exponents of powers with the
same base.
= 55
= 3,125
Simplify.
b. x5 • x7 • y2 • y
x5 • x7 • y2 • y = x5 + 7 • y2 + 1
Add the exponents of powers with
the same base.
= x12y3
Simplify.
Quick Check
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify 3a3 • (–5a4).
3a3 • (–5a4) = 3 • (–5) • a3 • a4
Use the Commutative Property
of Multiplication.
= –15a3 + 4
Add the exponents.
= –15a7
Simplify.
Quick Check
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression.
a. (23)3
(23)3 = (2)3 • 3
Multiply the exponents.
= (2)9
Simplify the exponent.
= 512
Simplify.
b. (g5)4
(g5)4 = g5 • 4
= g20
Multiply the exponents.
Simplify the exponent.
Quick Check
4-7
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression.
1. 22 • 23
32
3. 4r 6s • 7r 3s5
28r 9s6
2. g2 • h2 • h4 • h
g2 • h7
4. –(22)5
–1,024
5. (v3)8
v24
4-7
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Mari can package 14 seashells in a box. She has 360
seashells. How many full boxes does she have? How many
shells are left over after the boxes are filled?
25; 10
4-8
Exponents and Division
PRE-ALGEBRA LESSON 4-8
(For help, go to Lesson 4-4.)
Write in simplest form.
x2
1.
x
2.
6xy
3.
9y
4ab2
4.
16b
y
y2
Check Skills You’ll Need
4-8
Exponents and Division
PRE-ALGEBRA LESSON 4-8
(For help, go to Lesson 4-4.)
Solutions
x2
x • x1
1. x = x 1 = x
1
1
2. y2 = y
=
y
y
y•y1
3. 6xy = 2 • 31• x • y 1 = 2x
3•3•y
9y
3
1
1
1
1
1
2
ab
4. 4ab = 2 • 2 • a • b • b = a • b =
4
16b
2 • 2 • 2 • 2 • b1
2•2
1
1
4-8
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
a.
412
48
412
48
b.
= 412 – 8
Subtract the exponents.
= 44
Simplify the exponent.
= 256
Simplify.
w18
w13
w18
= w18
w13
= w5
– 13
Subtract the exponents.
Simplify the exponent.
Quick Check
4-8
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
73
a. (–12)73
(–12)
(–12)73
73
=
(–12)
73
(–12)
= (–12)0
=1
b.
8s20
32s20
8s20
32s20
1
= 4 s0
1
= 4 •1
1
= 4
– 73
Subtract the exponents.
Simplify.
Subtract the exponents. Simplify
8
.
32
Simplify s0.
Multiply.
4-8
Quick Check
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
12
a. 614
6
612
= 612
614
– 14
Subtract the exponents.
= 6–2
1
62
1
= 36
=
Write with a positive exponent.
Simplify.
4
b. z15
z
z4 = z4 – 15
z15
Subtract the exponents.
= z–11
= 111
z
Write with a positive exponent.
4-8
Quick Check
Exponents and Division
PRE-ALGEBRA LESSON 4-8
a2b3
Write
without a fraction bar.
ab15
a2b3
2 – 1b3
=
a
15
ab
= ab–12
– 15
Use the rule for Dividing Powers with the
Same Base.
Subtract the exponents.
Quick Check
4-8
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
1.
75
73
2.
–15b5c3
60b3c2
1
– 4 b2c
49
14
3. Write 6n24 without a fraction bar.
3n
2n–10
4-8
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Estimate in years the age of someone who is one million
minutes old.
about 2 years
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
(For help, go to Lesson 4-7.)
Write each expression with a simple exponent.
1. 103 • 105
2. 107 • 109
3. 105 • 10–3
4. 10–6 • 103
Check Skills You’ll Need
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
(For help, go to Lesson 4-7.)
Solutions
1. 103 • 105 = 103+5
2. 107 • 109 = 107+9
= 108
3. 105 • 10–3 = 105–3
= 102
= 1016
4. 10–6 • 103 = 10–6+3
= 10–3
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
About 6,300,000 people visited the Eiffel Tower in the
year 2000. Write this number in scientific notation.
6,300,000
Move the decimal point to get a decimal greater than 1
but less than 10.
6 places
6.3
6.3  106
Drop the zeros after the 3.
You moved the decimal point 6 places. The number is large.
Use 6 as the exponent of 10.
Quick Check
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write 0.00037 in scientific notation.
0.00037
Move the decimal point to get a decimal greater than 1
but less than 10.
4 places
3.7
3.7  10–4
Drop the zeros before the 3.
You moved the decimal point 4 places. The number is small.
Use –4 as the exponent of 10.
Quick Check
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in standard notation.
a. 3.6  104
3.6000
36,000
b. 7.2  10–3
007.2
0.0072
Write zeros while moving the decimal point.
Rewrite in standard notation.
Write zeros while moving the decimal point.
Rewrite in standard notation.
Quick Check
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in scientific notation.
a. 0.107  1012
0.107  1012 = 1.07  10–1  1012
= 1.07  1011
Write 0.107 as
1.07  10–1.
Add the exponents.
b. 515.2  10–4
515.2  10–4 = 5.152  102  10–4
= 5.152  10–2
Write 515.2 as 5.152  102.
Add the exponents.
Quick Check
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Order 0.035  104, 710  10–1, and 0.69  102 from
least to greatest.
Write each number in scientific notation.
0.035  104
3.5  102
710  10–1
0.69  102
7.1  10
6.9  10
Order the powers of 10. Arrange the decimals with the same
power of 10 in order.
6.9  10
7.1  10
3.5  102
Write the original numbers in order.
0.69  102, 710  10–1, 0.035  104
4-9
Quick Check
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Multiply 4  10–6 and 7  109. Express the result in
scientific notation.
(4  10–6)(7  109) = 4  7  10–6  109
Use the Commutative
Property of
Multiplication.
= 28  10–6  109
Multiply 4 and 7.
= 28  103
Add the exponents.
= 2.8  101  103
Write 28 as 2.8  101.
= 2.8  104
Add the exponents.
Quick Check
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Quick Check
In chemistry, one mole of any element contains
approximately 6.02  1023 atoms. If each hydrogen atom weighs
approximately 1.67  10–27 kg, approximately how much does
one mole of hydrogen atoms weigh?
Multiply number of atoms by
(6.02  1023)(1.67  10–27)
weight of each.
= 6.02  1.67  1023  10–27
10.1  1023  10–27
Use the Commutative Property of
Multiplication.
Multiply 6.02 and 1.67.
= 10.1  10–4
Add the exponents.
= 1.01  101  10–4
Write 10.1 as 1.01  101.
= 1.01  10–3
Add the exponents.
One mole of hydrogen atoms weighs approximately 1.01  10–3 kg.
4-9
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in scientific notation.
1. 5,400,000
2. 0.0000867
5.4  106
8.67  10–5
Write each number in standard notation.
3. 3.45  106
3,450,000
4. 1.99  10–5
0.0000199
5. Order 7.2  105, 7.2  106, 7.02  106, and 7.1  10–6 from least
to greatest.
7.1  10–6, 7.2  105, 7.02  106, 7.2  106
6. Multiply 14  106 and 4  10–4. Express the result in scientific
notation.
5.6  103
4-9