Download Ch 4, Factors and Exponents

Divisibility and Factors
Lesson 4-1
Pre-Algebra
Objectives: 1. to use divisibility tests
2. to find factors
Divisibility and Factors
Lesson 4-1
Pre-Algebra
New Terms: 1.
2.
Tips:
divisible – one integer is divisible by another if the remainder is 0
when you divide
factor – one integer is a factor of another integer (not zero) if it
divides that integer with a remainder zero.
the product of two integers is an integer, and both integers are factors
of the product. Moreover, both integers divide the product, and the
product is said to be divisible by each integer.
Divisibility and Factors
Lesson 4-1
Pre-Algebra
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.
Divisibility and Factors
Lesson 4-1
Pre-Algebra
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.
Divisibility and Factors
Lesson 4-1
Pre-Algebra
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.
Exponents
Lesson 4-2
Pre-Algebra
Objectives: 1. to use exponents
2. to use the order of operations with exponents
Exponents
Lesson 4-2
Pre-Algebra
New Terms: 1. exponents – you can use exponents to show repeated multiplication
2. power – a power has two parts, a base and an exponent
Tips:
the exponent is placed to the upper right of the base, and it only
applies to that base. If an exponent has as its base an expression,
that expression must be written in parentheses.
Exponents
Lesson 4-2
Pre-Algebra
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.
Exponents
Lesson 4-2
Pre-Algebra
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.
The star is 10,000 light-years from Earth.
Exponents
Lesson 4-2
Pre-Algebra
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.
Prime Factorization and Greatest Common Factor
Lesson 4-3
Pre-Algebra
Objectives: 1. to find the prime factorization of a number
2. to find the greatest common factor (GCF) of two or more numbers
New Terms: 1.
prime number – is an integer greater than 1 with exactly two
positive factors, 1 and itself
2.
composite number – is an integer greater than 1 with more than
two positive factors
3.
greatest common factor – factors that are the same for two or more
numbers or expressions are common factors. The greatest of
these is called the GCF.
Tips:
remember a factor is a number that divides evenly into another
number with a remainder of zero.
Prime Factorization and Greatest Common Factor
Lesson 4-3
Pre-Algebra
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.
Prime Factorization and Greatest Common Factor
Lesson 4-3
Pre-Algebra
Use a factor tree to write the prime factorization of
273.
273
Prime
Prime
3 • 7 • 13
273 = 3 • 7 • 13
3
•
91
7
•
Start with a prime factor.
Continue branching.
13
Stop when all factors are
prime.
Write the prime factorization.
Prime Factorization and Greatest Common Factor
Lesson 4-3
Pre-Algebra
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.
Simplifying Fractions
Lesson 4-4
Pre-Algebra
Objectives: 1. to find equivalent fractions
2. to write fractions in simplest for
New Terms: 1.
Tips:
simplest form – a fraction is in simplest form when the numerator
and the denominator have no common factors other than 1.
always check your answers and the steps involved in finding the answers
Simplifying Fractions
Lesson 4-4
Pre-Algebra
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
Simplifying Fractions
Lesson 4-4
Pre-Algebra
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
Simplifying Fractions
Lesson 4-4
Pre-Algebra
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.
Simplifying Fractions
Lesson 4-4
Pre-Algebra
(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
Problem Solving Strategy: Solve a Simpler Problem
Lesson 4-5
Additional Examples
Pre-Algebra
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?
Problem Solving Strategy: Solve a Simpler Problem
Lesson 4-5
Pre-Algebra
Additional Examples
(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.
Rational Numbers
Lesson 4-6
Pre-Algebra
Objectives: 1. to identify and graph rational numbers
2. to evaluate fractions containing variables
New Terms: 1.
Tips:
rational number – is any number you can write as a fraction
the quotient of two integers with the same sign is positive
Rational Numbers
Lesson 4-6
Pre-Algebra
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
Rational Numbers
Lesson 4-6
Additional Examples
Graph each rational number on a number line.
3
a. – 4
b. 0.5
c. 0
d. 1
3
Pre-Algebra
Rational Numbers
Lesson 4-6
Pre-Algebra
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.
Exponents and Multiplication
Lesson 4-7
Pre-Algebra
Objectives: 1. to multiply powers with the same base
2. to find a power of a power
Exponents and Multiplication
Lesson 4-7
Pre-Algebra
Tips:
when in doubt, write it out.
Exponents and Multiplication
Lesson 4-7
Pre-Algebra
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
= x12y3
Add the exponents of powers with
the same base.
Simplify.
Exponents and Multiplication
Lesson 4-7
Pre-Algebra
Simplify 3a3 • (–5a4).
3a3 • (–5a4) = 3 • (–5) • a3 • a4
Use the Commutative Property
of Multiplication.
= –15a3 + 4
Add the exponents.
= –15a7
Simplify.
Exponents and Multiplication
Lesson 4-7
Pre-Algebra
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.
Exponents and Division
Lesson 4-8
Pre-Algebra
Objectives:
1. To divide expressions containing
exponents
2. To simplify expressions with integer
exponents
Exponents and Division
Lesson 4-8
Pre-Algebra
Tips:
Sometimes students think an expression with a negative exponent
makes the expression negative, that is not true. A negative exponent makes the
number smaller (unless the base is less than one but greater than zero).
Exponents and Division
Lesson 4-8
Pre-Algebra
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.
Exponents and Division
Lesson 4-8
Pre-Algebra
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
Simplify s0.
Multiply.
8
.
32
Exponents and Division
Lesson 4-8
Pre-Algebra
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.
Exponents and Division
Lesson 4-8
Pre-Algebra
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.
Scientific Notation
Lesson 4-9
Pre-Algebra
Objectives: 1. to write and evaluate numbers in scientific notation
2. to calculate with scientific notation
New Terms: 1.
2.
Tips:
scientific notation – a way to write numbers using powers of 10
standard notation – write the number using all digit and decimal places
be careful with which way you move the decimal, think if the number is suppose to be smaller or larger
Scientific Notation
Lesson 4-9
Pre-Algebra
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 x 106
Drop the zeros after the 3.
You moved the decimal point 6 places. The original number is
greater than 10. Use 6 as the exponent of 10.
Scientific Notation
Lesson 4-9
Pre-Algebra
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 x 10–4
Drop the zeros before the 3.
You moved the decimal point 4 places. The original
number is less than 1. Use –4 as the exponent of 10.
Scientific Notation
Lesson 4-9
Pre-Algebra
Write each number in standard notation.
a. 3.6 x 104
3.6000
36,000
b. 7.2 x 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.
Scientific Notation
Lesson 4-9
Pre-Algebra
Write each number in scientific notation.
a. 0.107 x 1012
0.107 x 1012 = 1.07 x 10–1 x 1012
= 1.07 x 1011
Write 0.107 as
1.07  10–1.
Add the exponents.
b. 515.2 x 10–4
515.2 x 10–4 = 5.152 x 102 x 10–4
= 5.152 x 10–2
Write 515.2 as 5.152  102.
Add the exponents.
Scientific Notation
Lesson 4-9
Pre-Algebra
Order 0.035 x 104, 710 x 10–1, and 0.69 x 102 from
least to greatest.
Write each number in scientific notation.
0.035 x 104
3.5 x 102
710 x 10–1
0.69 x 102
7.1 x 10
6.9 x 10
Order the powers of 10. Arrange the decimals with the same
power of 10 in order.
6.9 x 10
7.1 x 10
Write the original numbers in order.
0.69 x 102, 710 x 10–1, 0.035 x 104
3.5 x 102
Scientific Notation
Lesson 4-9
Pre-Algebra
Multiply 4 x 10–6 and 7 x 109. Express the result in
scientific notation.
(4 x 10–6)(7 x 109) = 4 x 7 x 10–6 x 109
Use the Commutative
Property of
Multiplication.
= 28 x 10–6 x 109
Multiply 4 and 7.
= 28 x 103
Add the exponents.
= 2.8 x 101 x 103
Write 28 as 2.8  101.
= 2.8 x 104
Add the exponents.
Scientific Notation
Lesson 4-9
Pre-Algebra
In chemistry, one mole of any element contains
approximately 6.02 x 1023 atoms. If each hydrogen atom weighs
approximately 1.67 x 10–27 kg, approximately how much does
one mole of hydrogen atoms weigh?
Multiply number of atoms by
(6.02 x 1023)(1.67 x 10–27)
weight of each.
= 6.02 x 1.67 x 1023 x 10–27
10.1 x 1023 x 10–27
Use the Commutative Property of
Multiplication.
Multiply 6.02 and 1.67.
= 10.1 x 10–4
Add the exponents.
= 1.01 x 101 x 10–4
Write 10.1 as 1.01  101.
= 1.01 x 10–3
Add the exponents.
One mole of hydrogen atoms weighs approximately 1.01 x 10–3 kg.