Download Numbers - Concepts _ Properties Unit

Numbers: Concepts
& Properties
Exponents
 Raising a number/variable to a power
 Multiplying a number/variable by itself a
certain amount of times
 xn  x to the nth power
 x · x · x · · · (n copies of x)
 xn can be written as a product of n
amount of x’s
 “x” in this example is called the base, and
“n” is this example is called the power
Writing Exponents Multiple Ways
 Exponents can be represented in
multiple ways
1. normal exponential form  25
2. expanded form  2 · 2 · 2 · 2 · 2
3. using words  two to the fifth
power or a product of five copies of
two
You Try
1. Write 3 · 3 · 3 in exponential
form
3
2. Write 2 in expanded form
3. Write 4 · 4 · x · x in
exponential form
4. Write 5x4 in expanded form
Perfect Squares
 What is a perfect square?
- a perfect square is a number produced by
another number multiplied by itself
- for example, 12 = 1, so 1 is a perfect square; 22 =
4, so 4 is a perfect square; 32 = 9, so 9 is a perfect
square
- the area of square is always a perfect square if
the length of its side is a whole number
- list of perfect squares from 1 to 400: 1, 4, 9, 16,
25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,
256, 289, 324, 361, 400
Primes, Factors, and
Factorization
 What is a prime number?
- prime numbers are numbers that are only
divisible by 1 and itself
- this means that the only factors of a prime
number are 1 and itself
- if a number is not a prime number, then it is a
composite number
- list of prime numbers from 1 to 100: 1, 2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97
- the only even prime number is 2
Factors
 What are factors?
- in life, factors are things that cause something
to happen
- in math, factors are numbers that produce
another number when multiplied by each other
- examples of factors: the factors of 10 are 1, 2,
5, 10
- these numbers are factors of 10 because 10 is
produced when certain pairs of these numbers
are multiplied
Strategy for Finding
Factors
1. Know that 1 and the number itself are
always factors of the number
2. If the number is even, then 2 times another
number will produce that number
3. Check to see if the number is divisible by 3,
then 4, then 5, etc.
Example
List positive integer factors of 24.
Solution: 1, 24 since 1 x 24 = 24
2, 12 since 2 x 12 = 24
3, 8 since 3 x 8 = 24
4, 6 since 4 x 6 = 24
You try
1. List all of the positive integer
factors of 18
2. List all of the positive integer
factors of 21
Homework
Do problems 1 – 22 on NCP
Practice Problems Part I
sheet
Prime Factorization
 Creating a factor tree
1. Find a pair of factors
2. Factor each number until
only prime numbers remain
3. Circle all primes
Example 1
 Create a factor tree for
24
24
2
12
2
6
2
3
Example 2
Create a factor tree for 18
18
2
9
3
3
Example 3
Create a factor tree for
30x2y3
30x2y3
x2
6
5
2
3
x
y3
x
y
y
y
You Try
1. Create a factor tree for 36
2. Create a factor tree for 39
3. Create a factor tree for 50xy2
Final Step For Prime
Factorization
 Once we have created a factor tree,
list all of the prime factors as a
product in expanded form and then
put in normal exponential form
- in example 1, 24 = 2 ·2 · 2 · 3 = 23 · 3
- in example 2, 18 = 2 · 3 · 3 = 2 · 3²
- in example 3, 30x²y³ = 2 · 3 · 5 · x · x ·
y ·y · y = 2 · 3 · 5 · x² · y³
You Try
1. Prime Factorize 45
2. Prime Factorize 60
Homework
Do problems 23 – 25 on NCP
Practice Problems sheet and
problems 1 – 5 on additional
problems sheet.
Greatest Common Factor (GCF)
 What is greatest common factor (gcf)?
- GCF is the largest number that goes into 2
or more numbers
- Example 1: Find the GCF between 18 and 20
a. List all of the factors of 18 and 20
18: 1, 2, 3, 6, 9, 18
20: 1, 2, 4, 5, 10, 20
b. Identify the largest number that is in
both lists: 2  the GCF between 18 and 20
Another Way to Find the GCF
1. Prime factorize all numbers
2. List all primes they have in common
3. Identify the lowest power in each of
the primes they have in common
4. Multiply the primes along with the
power to get the GCF
Examples
Example 1: Find the GCF between 45 and 60
Step 1 – 45 = 32 · 5 and 60 = 22 · 3 · 5
Step 2 – 3 and 5 are the primes in common
Step 3 – 31 and 51 are the primes with the
lowest powers in common
Step 4 – 31 · 51 = 15
So, 15 is the GCF between 45 and 60.
Another Example
Example 3: Find the GCF between 54 and 96
Step 1 – 54 = 33 · 2 and 96 = 25 · 3
Step 2 – 2 and 3 are the primes in common
Step 3 – 21 and 31 are the primes in common
with the lowest powers
Step 4 – 2 · 3 = 6
So, 6 is the GCG between 54 and 96.
You Try
1. Use prime factorization to find the GCF
between 20 and 48
2. Use prime factorization to find the GCF
between 60 and 200
Using the GCF to Reduce
Fractions
Example: Reduce 72/120 to lowest terms
Step 1 – 72 = 23 · 32 and 120 = 23 · 3 · 5
Step 2 – 23 · 3 = 24 is the GCF between 72 and
120
Step 3 – 72 ÷ 24/120 ÷ 24 = 3/5
So, 3/5 is 72/120 in lowest terms.
Note: To reduce a fraction to lowest terms, (1) find
the GCF between the numerator and the
denominator and then (2) divide the numerator
and denominator by the GCF.
Another Way to Reduce a
Fraction to Lowest Terms
Step 1 - Prime factorize numerator and
denominator but leave in expanded form
example: 72 = 2 · 2 · 2 · 3 · 3 and 120 = 2 · 2 · 2 · 3 ·
5
Step 2 – Cross out (cancel) the same numbers in the
numerator and denominator
example: 2 · 2 · 2 · 3 · 3/2 · 2 · 2 · 3 · 5
Step 3 – Multiply what remains in the numerator
and multiply what remains in the denominator
for the reduced fraction
example: 3/5  so, 72/120 reduces to 3/5
You Try
1. Reduce 180/600 to lowest terms
Homework
Do all of the problems on NCP
Practice Problems Part II and the
additional GCF practice problems.
Lowest Common Multiple (LCM)
 What is lowest common multiple (LCM)?
- LCM is the smallest multiple that two or
more numbers have in common
 What is a multiple?
- A multiple is a number that is produced by
another number when multiplied
- Factors are the parents and multiples are the
offspring
One Way to Find the LCM
Example 1: Find the LCM between 4 and 6
Step 1 – list some multiples of 4 and 6
example: 4 – 4, 8, 12, 16, 20, 24
6 – 6, 12, 18, 24, 30
Step 2 – identify the first multiple that they
have in common which is the LCM
example: 12  the LCM between 4 and 6
Another Way to Find the LCM
Step 1 – prime factorize both number
example: 4 = 22 and 6 = 2 · 3
Step 2 – write down all of the primes with the
highest power of those primes
example: 2² and 3
Step 3 – multiply these numbers to get the LCM
example: 2² · 3 = 12
More Examples
Example 2: find the LCM between 6 and 8
Step 1 – 6 = 2 · 3 and 8 = 2³
Step 2 – 2³ · 3 = 24
So, 24 is the LCM between 6 and 8.
Example 3: find the LCM between 25 and 45
Step 1 – 45 = 3² · 5 and 25 = 5²
Step 2 – 3² · 5² = 225
So, 225 is the LCM between 25 and 45.
You Try
1. Find the LCM between 18 and 50
2. Find the LCM between 21 and 35
Using LCM to Order Fractions
Example 1: Order the following fractions from least to greatest by
finding the LCM: 9/21, 7/15, and 18/40
Step 1 – find the LCM between all denominators
example: 21 = 3 · 7, 15 = 3 · 5, and 40 = 2³ · 5
LCM = 3 · 5 · 7 · 2³ = 840
Step 2 – rewrite all of the fractions with the LCM as the denominator
by multiplying numerator and denominator by the same factor
example: 9/21 = 9 · 40/21 · 40 = 360/840
7/15 = 7 · 56/15 · 56 = 392/840
18/40 = 18 · 21/40 · 21 = 378/840
So, the following is the order of fractions from least to greatest:
9/21, 18/40, and 7/15
You Try
Order the following fractions from least to
greatest by finding the LCM: 13/24, 27/50, and
55/108
Homework
Do all problems on NCP Practice
Problems Part III
Radicals
 What are radicals?
- Radicals are roots of numbers such
as square roots, cube roots, fourth
roots, etc.
- 24 is stated as “radical 24” or
“the square root of 24”
- this means “what number
multiplied by itself will produce 24?”
Components of Radicals
a
c b
radicand
coefficient
index
 The radicand is the number inside of the radical
 The index tells us what the root is. For example, if no
number is there, then it is a square root; if a three is
there, then it is a cube root, and so on.
 The coefficient is always outside of the radical and is
being multiplied by the radical. If no number is there,
then the coefficient is 1.
Simplifying Radicals
 Approximation Method
 - in your calculator hit “shift,” “x2,”
type in the radicand, and then hit
“EXE or enter”
 - example:
24 ≈ 4.90
Other Way To Simplify
Radicals
 Exact Method
Step 1 – prime factorize the radicand
example: 24 = 2³ · 3
Step 2 – if the radical is a square root, take
all of the squares out of the radical, put
the base of the squares as coefficients,
and leave what remains in the radicand;
example: 2 2 · 3 = 2 6
Another Example
Example 2: Simplify
Solution:
28
28
=
=
2² · 7
2 7
Example 3: Simplify 34
34 = 17 · 2
Solution:
= 34
Since there are no squares inside the radical,
then the radical is already in simplified form.
You Try
1. Simplify
21
2. Simplify
42
3. Simplify
216
4. Simplify
60
Homework
Do problems 1 – 7 on NCP Practice Problems
Part IV sheet and all of the problems on the
“Simplifying Radicals” Additional Practice
Problems sheet.
Adding and Subtracting
Radicals
What are “like” radicals?
- “Like” radicals have the same radicand
Radicals can only be added or subtracted if
they are “like”
When adding/subtracting radicals, only
add/subtract the coefficients
Remember: if there is no coefficient, then
the coefficient is 1
Examples
Example 1:
10  4 10  (1  4) 10  5 10
Example 2: 2 5 - 6 5  (2 - 6) 5  - 4 5
You Try
1.
7 3 5 3
2.
2 8 2
When Radicals Are Not “Like”
 If radicals are not “like” but can be
simplified, simplify first and then check for
“like” radicals.
- if the simplified radicals are “like,” then
add/subtract
- if simplified radicals are not “like”, then
stop
Examples
Example 1: 10 2  2 8
Solution:
2 84 2
10 2  4 2  6 2
More Examples
Example 2:
Solution:
5 12  2 15
5 12  10 3
10 3  2 15
The answer cannot be simplified any further.
You Try
1.
6 12  4 3
2.
5 5  3 30
Homework
Do problems 8 – 11 on NCP Practice
Problems Part IV sheet and all of
the problems on “Adding and
Subtracting Radicals Additional
Practice Problems” sheet.
Multiplying Radicals
 When multiplying radicals, multiply the
radicands by each other and multiply the
coefficients by each other. Then simplify the
radicals if possible.
 If multiplying a number by a radical, then
multiply that number by the coefficient of the
radical
Examples
Example 1:
3 5 (2 6 )  3  2 5  6  6 30
Example 2:
6 7 (4 8 )  6  4 7  8  24 56  24
Example 3:
2 14  24  2
2
14  48 14
8  2 5  16 5
Example 4:
2 10  2  2 1 20  2 20  2
2 5  2 2
2
54 5
You Try
1.
10 3 (7 6 )
2.
11 5 (2 10 )
Multiplying Radicals:
Monomials by Binomials
 Multiply the radical that is outside of the
parentheses by both of the radicals inside of
the parentheses. Then add the products.
Example 1:
 2 10 (4 5  2)  2 10  4 5  2 10  2  8 50  4 10
 8 50  4 10  40 2  4 10
You Try
1.
3 (4  6 )
Homework
Do problems 12 – 15 on NCP Practice Problems
Part IV sheet. Also, do all of the problems on
the “Multiplying Radicals Additional Practice
Problems” sheet.
Dividing Radicals
 Radicals cannot be in the denominator of
fractions
 To get radicals out of denominator, multiply
numerator and denominator by radical in
denominator. Then simplify radical in
numerator if possible. Then reduce fraction if
possible.
Examples
Example 1:
2
2 5
10
10




5
5
5 5
25
Example 2:
3
3
6
18
18 3 2
2






12
4
2 6 2 6 6 2 36 2  6
You Try
1.
6
2
2.
3 5
2 10
Homework
Do problems 16 and 17 on NCP Practice
Problems Part IV. Also, do all of the problems
on the “Dividing Radicals Additional Practice
Problems” sheet.
Properties of Exponents
x x  x
a
1.
b
a b
When multiplying 2 exponents with
the same base, keep the base the
same and add the powers.
3
4
7
Example 1:
 
x x x
More Properties
2. For every x≠ 0,
a
x x
x
a b
b
When dividing exponents with the same base,
keep the base the same and subtract the
powers as long as x does not equal 0.
Example 2: 7
x 2  x5
x
More Properties
b
ab
3.
(x )  x
When raising a power to a power,
multiply the powers.
4
Example 3:
8
2
a
(x )  x
More Properties
4. For every x ≠ 0,
x
a

1
x
a
Raising a number to a negative power is the
same as 1 over the number being raised to
the positive power as long as x does not 0.
Example 4:
1
6
x

x
6
More Properties
5.
x 1
Any number raised to the 0 power is 1.
0
Example 5:
0
4
1
You Try
Your answers should only contain positive
exponents.
8
3
1.
aa
2
aa
5
4
2.
3.
4
5
a
( 2 3)
aa
3
a
( 1)
a
Homework
Do problems 1 – 12 on NCP Practice
Problems Part V sheet.