Download Module 3 Notes Prime numbers: -Prime numbers have exactly two

Module 3 Notes
Prime numbers:
-Prime numbers have exactly two factors, which are 1 and itself. Example: 2, 3, 5,
7, 9, 11
Note: The numbers 0 and 1 are not prime numbers, and they do not have any prime
factors.
Prime Factorization
-The prime factorization of a number is the expression of a number as the product of
its prime factors. One way to demonstrate this is to use a factor tree. For example,
the prime factorization of 120 can be determined by breaking 120 into factors until
all the factors are prime numbers.
-Notice that 120 is the product of 10 x 12.
-Factor 10 into 2 x 5, and factor 12 into 4 x 3.
-Because 2, 5, and 3 are all prime, they are highlighted in red.
-The number 4 is still not prime, so factor it into 2 x 2.
-Therefore, the prime factorization of 120 is 2 x 2 x 2 x 3 x 5.
Sometimes, more than one factor tree can be made for numbers. For example, 120
could also be broken into the following factor tree:
Lowest Common Multiple
-The LCM is the acronym for the lowest common multiple of two or more
numbers.
-To find the LCM, perform the following:
1) List the several multiples of the smallest number.
2) List multiples of the larger number until a multiple in the first list is matched.
(This might mean you must write more multiples in the first list.)
3) The smallest number that is common to both lists is the LCM.
Example: Find the lowest common multiple of 6 and 9.
Step1) 6 ->
6 (6x1), 12 (6x2), 18 (6x3), 24(6x4), 30(6x5), 36(6x6), ...
Step 2) 9->
9 (9x1), 18(9x2), 27(9x3), 36(9x4),...
Step 3) Although 6 and 9 have many common multiples, the lowest common
multiple is 18.
Greatest Common Factor
-The GCF (greatest common factor) is the largest factor shared by two or more
numbers. It is also the largest number that divides into two or more numbers
without a remainder.
Example: What is the greatest common factor of 24 and 36?
Step1) List all the factors of 24.
1 x 24, 2 x 12, 3 x 8, 4 x 6 …… The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and
24.
Step 2) List all the factors of 36.
1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6 ….. The factors of 36 are 1, 2, 3, 4, 6, 9,
12, 18, and 36.
Step 3) The greatest common factor is 12 because it is the largest number that
is common to both lists. Notice that the numbers have more than one factor in
common, but 12 is the largest common factor.
Number Systems
-Natural numbers (N) 
-Whole numbers (W) 
-Integers numbers (I) 
1, 2, 3, . . .
0, 1, 2, . . .
. . . 3, 2, 1, 0, 1, 2, 3, . . .
-Rational numbers (Q)  numbers that can be written as
written as a decimal the decimal terminates or repeats
a
b
but b≠0 and when
a
b
(Q)
-Irrational

numbers that can’t be written as and when written as a
decimal the decimal doesn’t terminates nor does it repeat; best example is  .
-Real Numbers  the union of rational and irrational numbers. If the number
exists, it is a Real Number.
Q
I
W
N
Q
=
Reals
-The chart above shows that if you are a natural number then you are also a whole
number, an integer, and a rational number.
-If you are an integer you are also a rational number but you may not be a whole
number nor a natural number.
Ordering Real Numbers
-Real numbers can be plotted as points on a number line. A number line is a visual
representation of real numbers. The numbers on the number line increase from left
to right and have an origin at the point 0.
Example
Radicals
-The
symbol is called a radical sign, and is used to indicate the square root of
the expression under the radical sign. Thus,
= 3 and –3. This lesson is
concerned only with the principal (positive) square root.
-The
symbol has the same meaning as
, but most textbooks and math
literature do not include the little 2 that indicates the index. If no index is shown, it
is assumed to be 2 that signifies square root. The radical is made up of the following
components:
For example
, has a radicand of 9 and index of 2.
Examples: Identify the index and radicand for the following:
a. Index: 2; Radicand: 13
b. Index: 2; Radicand: 3
c. Index: 7; Radicand: 28
Perfect Squares
-A perfect square is a number that can be expressed as the product of two equal
factors. The two equal factors also represent the square root of the perfect square.
For example, the following numbers are all perfect squares:
-1 is a perfect square because it is the product of 1 x 1; therefore, 1 is the square
root.
-4 is a perfect square because it is the product of 2 x 2; therefore, 2 is the square
root.
-9 is a perfect square because it is the product of 3 x 3; therefore, 3 is the square
root.
-16 is a perfect square because it is the product of 4 x 4; therefore 4 is the square
root.
Another way to think of perfect squares and square roots is to consider the side
length and area of a square.
For example, the following square has a side length of 5 units
The area of the square is 25. Can you see how the area of a square (25) and its side
length (5) are related to a perfect square (25) and its square root (5)?
Perfect Cubes
-Perfect cubes and cube roots are similar to perfect squares and square roots. The
difference is that a perfect cube is a number that is formed by mulitplying a number
by itself three times instead of twice.
Please note that when the index is 3 or greater, the index is always shown. An index
of 3 means the cube root.
The symbol for cube root is
.
For example, because 4 x 4 x 4 = 64, the number 64 is a perfect cube. The number
64 is formed by multiplying its cube root, 4, by itself three times.
The following table shows perfect cubes and their cube roots:
Cube Root
Meaning
1
1x1x1
2
2x2x2
3
3x3x3
4
4x4x4
5
5x5x5
You will notice that 1 is both a perfect square and a perfect cube.
Perfect Cube
1
8
27
64
125
Another way to look at cube roots is by examining the volume of a cube. The
following cube has a length, width and height that are each 6 units.
The volume of this cube is 216 cubic units. The cube of 6 is 216; therefore the cube
root of 216 is 6.
Entire and Mixed Radicals
-Numbers of the form
are called entire radicals; for example
are entire radicals.
-Numbers in the form
are called mixed radicals; for example
are mixed radicals. In the mixed radical, "a" is called the
coefficient.
As you have seen previously, any radical containing a radicand that is a perfect
power of the index is a rational number and can, therefore, be expressed as a nonradical.
Example:
This rule can be generalized as follows:
Exponent Laws
Product of Product Law
Power of a power Law
Power of a product Law
Division of powers Law
Quotient Law
Negative exponent Law
Zero exponent Law
Rational exponents
-Rational exponents are exponents that are fractions. If an expression has a rational
exponent, it can be re-written as a radical expression.
Writing Rational Exponents
Examples: Write the following radicals in exponential form.