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.

26 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 64

exponent

power

base

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.

34 = 3 ∙ 3 ∙ 3 ∙ 3 4 is the exponent

Using Order of Operations with Exponents

1. Work inside grouping symbols2. Simplify any terms with exponents3. Multiply and divide in order from left to

right4. 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 the number.

• You can use a factor tree to find prime factorizations.

• Write the final factors in increasing order from left to right.

• Use exponents to indicate repeated factors.

825

5 165

5 33

3 11

5 ∙ 5 ∙ 3 ∙ 11 Write the prime factorization

prime

prime

prime

Start with a prime number

Continue branching

Stop when all factors are prime

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 12

2

2 2

4 3 4

2 2

2 ∙ 2 = 4

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 1 + 9 + 2 = 12 Simplify the denominator 2 – 5 = - 3

• To simplify a fraction with variables, first substitute for the variables.

a + b = 6 + - 5 – 3 = - 3 a = 6 b = - 5

= - 4simplest

form

= - 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 Algebra

23 ∙ 24 = 23+4 = 27 am ∙ an = am+n, for positive

integers m and n.

Using the Commutative Property

• Simplify -2x2 ∙ 3x5

-2x2 ∙ 3x5 = -2 ∙ 3 ∙ x2 ∙ x5

= -6x2+5

= -6x7

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)

= 72 + 2 + 2

= 76

• Notice that (72)3 = 76 = 72 ∙ 3. You can raise a power to a power by multiplying the exponents.

Use 72 as a base 3 times

When multiplying powers with the same base, add the exponents.

Simplify

Key Concept: Finding a Power

• To find a power of a power, multiply the exponents.

Arithmetic Algebra

(23)4 = 23 ∙ 4 = 212 (am)n = am ∙ n, 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.

78 = 7 · 7 · 7 · 7 · 7 · 7 · 7 · 7

73 7 · 7 · 7

= 7 · 7 · 7 · 7 · 7 · 7 · 7 · 7 7 · 7 · 7 = 75

= 78 = 75 = 78-3

73

Expand the numeratorand denominator

111

1 1 1

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:

50 = 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.

56

58= 5 6 - 8

= 5 - 2

= 1 52

= 1 25

Subtract the exponents

Write with a positive exponent

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 side. For instance, "x–2" just means "x2, but underneath, as in 1/(x2)".

• 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.

7,500,000,000,000 = 7.5 x 1012

Second factor is a power of 10

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-common-factor.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