Transcript Chapter 4

Chapter 4
Factors, Fractions, and
Exponents
Section 4-1
Divisibility and Factors
Divisibility and Factors
• One integer is divisible by another if the
remainder is 0 when you divide.
• One integer is a factor of another
nonzero integer if it divides that integer
with remainder zero.
Divisibility Rules for 2, 5, and 10
A number is divisible by
• 2 if the last digit is 0, 2, 4, 6 or 8
• 5 if the last digit is either 0 or 5
• 10 if the last digit is 0
Even numbers end in 0, 2, 4, 6, or 8 and are
divisible by 2.
Odd numbers end in 1, 3, 5, 7, or 9 and are
not divisible by 2.
Divisibility Rules for 3 and 9
A number is divisible by
• 3 if the sum of its digits is divisible by 3
• 9 if the sum of its digits is divisible by 9
Divisibility Rules for 4, 6, and 8
A number is divisible by
• 4 if the number formed by the last two
digits is divisible by 4
• 6 if it is divisible by 2 AND it is divisible
by 3
• 8 if the number formed by the last three
digits is divisible by 8
Section 4-2
Exponents
Using Exponents
• You can use exponents to show
repeated multiplication.
exponent
base
6
2 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 64
power
The base 2 is used as a
factor 6 times.
The value of
the
expression
Exponents
• A power has two parts
– A base is the repeated factor of a number
written in exponential form.
54 = 3 ∙ 3 ∙ 3 ∙ 3
5 is the base
– An exponent is a number that show how
many times a base is used as a factor.
4
3 =3∙3∙3∙3
4 is the exponent
Using Order of Operations
with Exponents
1. Work inside grouping symbols
2. Simplify any terms with exponents
3. Multiply and divide in order from left to
right
4. Add and subtract in order from left to
right
Section 4-3
Prime Factorization and
Greatest Common Factor
Prime or Composite?
• A prime number is an integer greater than 1 with
exactly two positive factors, 1 and the number itself.
– The numbers 2, 3, 5, and 7 are prime numbers.
• A composite number is an integer greater than 1 with
more than two positive factors.
– The numbers 4, 6, 8, 9, and 10 are composite numbers.
– The number 1 is neither prime nor composite.
Prime or Composite Numbers
Prime Factorization
• Writing a composite number as a
product of its prime factors
shows the prime factorization of
prime
the number.
• You can use a factor tree to find
prime factorizations.
5
prime
• Write the final factors in
increasing order from left to right.
• Use exponents to indicate
repeated factors.
825
5 ∙ 5 ∙ 3 ∙ 11
prime
Start with a
prime number
165
5
Continue
branching
33
3
11
Stop
when all
factors
are prime
Write the prime factorization
825 = 3 ∙ 52 ∙ 11 Use exponents to write the prime factorization
Finding the Greatest Common Factor
• Factors that are the same for two or more
numbers or expressions are common factors.
• The greatest of these common factors is
called the greatest common factor (GCF).
• You can use prime factorization to find the
GCF of two or more numbers or expressions.
• If there are no prime factors and variable
factors in common, the GCF is 1.
How to Find the GCF
• Let's use 36 and 54 to find their greatest common
multiple.
• The prime factorization of 36 is 2 x 2 x 3 x 3
• The prime factorization of 54 is 2 x 3 x 3 x 3
• Notice that the prime factorizations of 36 and 54 both
have one 2 and two 3s in common. So, we simply
multiply these common prime factors to find the
greatest common factor.
Like this...
2 x 3 x 3 = 18
Section 4-4
Simplifying Fractions
Reducing Fractions to Lowest
Terms
• A fraction is in its simplest form (this is
also called being expressed in lowest
terms) if the Greatest Common Factor
(GCF), also called the Greatest
Common Divisor (GCD), of the
numerator and denominator is 1. For
example, 1/2 is in lowest terms but 2/4
is not.
Finding Equivalent Fractions
• Equivalent fractions are different fractions that
are equal to the same number and can be
simplified and written as the same fraction
– For example, 3/6 = 2/4 = 1/2 and 3/9 = 2/6 = 1/3).
• Equivalent fractions describe the same part of
a whole.
• You can find equivalent fractions by
multiplying or dividing the numerator and
denominator by the same nonzero factor.
Two Methods to Simplifying
Fractions
Method 1
• Try dividing both the top and bottom of
the fraction until you can't go any further
(try dividing by 2,3,5,7,... etc).
Example: Simplify the fraction 24/108 :
Method 2
• Divide both the top and bottom of the fraction
by the Greatest Common Factor, (you have to
work it out first!).
• Example: Simplify the fraction 8/12 :
– The largest number that goes exactly into both 8
and 12 is 4, so the Greatest Common Factor is 4.
– Divide both top and bottom
by 4:
8
2
12
4
2
2∙2=4
3
2
4
2
2
So the largest
number that
goes into
both 8 and 12
is 4.
Writing Fractions in Simplest
Form
To simplify a fraction, you should follow four steps:
• Write the prime factorization of both the numerator and
denominator. (The process for finding prime factors was
explained in the previous section).
• Rewrite the fraction so that the numerator and
denominator are written as the product of their prime
factors.
• Cancel out any common prime factors.
• Multiply together any remaining factors in the numerator
and denominator.
http://cstl.syr.edu/fipse/fractions/Unit2/Unit2c.html
Section 4-6
Rational Numbers
Integers
• Integers are the whole numbers, negative
whole numbers, and zero. For example,
43434235, 28, 2, 0, -28, and -3030 are
integers, but numbers like 1/2, 4.00032, 2.5,
, and -9.90 are not.
• It is often useful to think of the integers as
points along a 'number line', like this:
Note that zero is neither positive nor negative.
Rational Numbers
• A rational number is any number that can be
written as a ratio of two integers (hence the
name!). In other words, a number is rational if
we can write it as a fraction where the
numerator and denominator are both
integers.
• So the set of all rational numbers will contain
the numbers 4/5, -8, 1.75 (which is 7/4), 97/3, and so on.
Identifying and Graphing
Rational Numbers
• A rational number is any number you
can write as a quotient a/b of two
integers where b is not zero.
• All integers are rational numbers. This
is true because you can write any
integer a as a/1.
Writing Equivalent Fractions
There are two basic methods that we use:
• We can multiply both numerator and denominator by
the same number, and we will create a new fraction
equivalent to the original one;
• We can divide both numerator and denominator by
the same number, and we will again create a new
fraction equivalent to the original one.
Evaluating Fractions Containing
Variables
• Recall that a fraction bar is a grouping symbol, so
you first simplify the numerator and the denominator.
Then, simplify the fraction.
Simplify the numerator
Simplify the denominator
1 + 9 + 2 = 12
2–5 = -3
= -4
simplest
form
• To simplify a fraction with variables, first substitute for
the variables.
a+b = 6+-5
–3 = -3
a=6
b=-5
= -1
3
Section 4-7
Exponents and
Multiplication
Multiplying Powers with the
Same Base
• To multiply numbers or variables with
the same base, add the exponents.
Arithmetic
3
4
3+4
2 ∙2 =2
Algebra
7
=2
m
n
m+n
a ∙a =a
, for
positive integers m and n.
Using the Commutative Property
2
• Simplify -2x ∙ 3x
2
5
5
2
-2x ∙ 3x = -2 ∙ 3 ∙ x ∙ x
= -6x
2+5
= -6x
7
5 Use the Commutative Property
of Multiplication
Add the exponents
Simplify
Finding a Power of a Power
• You can find the power of a power by using
the rule of Multiplying Powers with the Same
Base.
(72)3 = (72) ∙ (72) ∙ (72)
Use 72 as a base 3 times
When multiplying powers with the same
base, add the exponents.
2+2+2
=7
= 76
Simplify
2 3
6
2∙3
• Notice that (7 ) = 7 = 7 . You can raise a
power to a power by multiplying the
exponents.
Key Concept: Finding a Power
• To find a power of a power, multiply the
exponents.
Arithmetic
3 4
3∙4
(2 ) = 2
Algebra
12
=2
m n
m∙n
(a ) = a
, for
positive integers m and n.
Section 4-8
Exponents and Division
Dividing Expressions Containing
Exponents
•
To divide powers with the same base, you
subtract exponents.
8
7 =7·7·7·7·7·7·7·7
73
7·7·7
1
1
1
=7·7·7·7·7·7·7·7
1 1
1
7·7·7
5
=7
= 78 = 75 = 78-3
73
Expand the numerator
and denominator
Divide the common
factors
Dividing Powers
with the Same Base
• To divide numbers or variables with the
same nonzero base, subtract the
exponents.
Zero as an Exponent
• The 'Zero Exponent' rule is really easy, but you
will have to memorize it, because it does not
seem to make sense! Here it is:
0
5 =1
• Any power of zero always
equals 1.
• But consider what it really means:
"When you multiply 5 by itself
NO times, you get 1.
Positive Exponents
Simplify each expression.
6
5
6-8
Subtract the exponents
=
5
8
5
-2
Write with a positive exponent
=5
=1
2
5
=1
25
Simplify
Negative Exponents
• A negative exponent just means that the
base is on the wrong side of the fraction line,
so you need to flip the base to the other
–2
2
side. For instance, "x " just means "x , but
2
underneath, as in 1/(x )".
• A negative exponent is equivalent to the
inverse of the same number with a positive
exponent. In other words:
Section 4-9
Scientific Notation
Scientific Notation
• Scientific notation is used to express very large or very
small numbers.
• It provides a way to write numbers using powers of 10.
– You write a number in scientific notation as the product of two
factors.
– A number in scientific notation is written as the product of a number
(integer or decimal) and a power of 10.
• The number has one digit to the left of the decimal point.
The power of ten indicates how many places the decimal
point was moved.
Second factor is
a power of 10
7,500,000,000,000 = 7.5 x 10
12
First factor is greater than or equal
to 1, but less than 10
Scientific Notation Cont. . . .
• The number has one digit to the left of the decimal
point. The power of ten indicates how many places
the decimal point was moved.
• The decimal number 0.00000065 written in scientific
notation would be 6.5x10-7 because the decimal
point was moved 7 places to the right to form the
number 6.5.
• A decimal number smaller than 1 can be converted to
scientific notation by decreasing the power of ten by
one for each place the decimal point is moved to the
right.
Works Cited
• http://www.mathgoodies.com/lessons/vol3/divisibility.html
• http://www.helpwithfractions.com/greatest-commonfactor.html
• http://www.mathsisfun.com/simplifying-fractions.html
• http://cstl.syr.edu/fipse/fractions/Unit2/Unit2c.html
• http://www.enchantedlearning.com/math/fractions/reducing/
• http://mathforum.org/dr.math/faq/faq.integers.html
• http://www.algebra-online.com/equivalent-fractions-2.htm