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Powers and Exponents
Solve the following 8 x 4 x 2 =
What are each of the digits above called?
1. Evaluate: 2 x 2 x 2 x 2
What is another way we can write the example of number 2?
24
Vocabulary:
Factors-
_____________________________________________________________________________
Exponential Form-
_________________________________________________________________________
Base -
_____________________________________________________________________________
Exponent-
_____________________________________________________________________________
Write each of the following as a product of the same factor.
2. 106 6. 54
3. 62 7. 106
4. 41
Write in exponential form.
8. 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9
9. 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
Powers and Exponents
Evaluate:
10. 35 =
11. 43 =
12. 104 =
13. 107 =
**Hint-
If 81 is the product and the base is 3, what is the exponent?
13. 3? = 81
*Think- how many times do I need to multiply the base by itself to get 81?
Classwork:
Write each of the following as the product of the same factor.
1. 82 2. 97
3. 43 4. 14
5. 710 6. 1012
Write in exponential form.
7. 3 ∙ 3 ∙ 3 ∙ 3 8. 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5
9. 12 ∙ 12 ∙ 12 ∙ 12 ∙ 12 10. 6 ∙ 6 ∙ 6 ∙ 6 ∙ 6
Evaluate.
11. 105 12. 73
13. 85 14. 1010
Find the exponent in each of the following
15. 3? = 27 16. 4? = 64 17. 5? = 125 18. 6? = 1296
Order of Operations
Vocabulary
Numerical Expression- ________________________________________________________
Some mathematical expressions involve several operations. Does the order in which these
operations are done make a difference?
6 + 3 x 4 6 + 3 x 4
9 x 4 = 36 6 + 12 = 18 Which is correct, and why?
***Please Excuse My Dear Aunt Sally!
Solve each of the following below.
1. 1 + 7 x 3 2. (1 + 7) x 3
Do they have the same answer? Are parenthesis important?
Order of Operations
Evaluate each of the following numerical expressions.
3. 21 – (18 + 2) 4. 15 + 5 x 3 – 2 5. 32 – 3 x 7 + 4
6. 36 ÷ 9 x 2 7. 12 ÷ 24 8. 28 ÷ (3 – 1)
9. 1
4𝑥 (
1
5+
1
2) −
1
10 10. (
1
4)2𝑥
1
2−
1
10 11. (.25)2 +
1
2 x 2
12. . (3.4+4.4)×(4.8÷0.4) 13. (8.5−5.9)×(4.8÷1.6) 14. 18( ½) – 9
15. 6 + 52- 12 + 4 16. 5 x ( 2-1)2+ 17 17. (27 ÷ 3)2 – 3
Order of Operations
18. On his math test, Steve scored 5 points on each of 5 questions, 2 points on each of 2
questions and 3 points on each of 4 questions. To find the number of points he scored, evaluate
the expression below:
55 + 22 + (3 x 4)
19. Robert and Lily solved the expression below. Robert got an answer of 94 and Lily got an
answer of 64.
80 ÷ (2 x 1)3 + 54
a) Who is correct?
b) Explain why it is important to use the proper order of operations to solve.
20. 12 − (8
5+ 3 ÷
2
3 ) 21. 4 − (
4
3𝑥
5
4+
11
6 )
Properties
Properties of Operations
Commutative Property- numbers can be added or multiplied in any order to give you the same
answer.
Ex:
Associative Property- numbers are grouped differently to give you the same answer.
Ex:
Identity Property of Addition- when you add anything with zero is equals itself.
Ex:
Identity Property of Multiplication- when you multiply anything with one it equals itself.
Ex:
Zero Property- anything multiplied by zero gives you a product of zero
Ex:
Distributive Property- this is when you break down one of the numbers, multiply the factor’s
addends by a number, then add the products together.
Ex:
Using your notes, name the property for each below.
1. 32 x 1 = 32
2. 10 + 6+ 2 = 6 + 2 + 10
3. 8 x 14 = (8 x 14)+(8 x 4)
4. (3 x 5) x7 = 3x (5 x7)
5. 0 + 12 = 128 + 0 = 8
6. mt = tm
7. 11 + ( 4 + 8) = (11 + 4) +8
8. 5 x (9 + 7) = (5x9) + (5x7)
Properties
Properties of Addition
Property Example:
Commutative
Associative
Identity
Properties of Multiplication
Property Example:
Commutative
Associative
Identity
Zero
Distributive
Find the missing number in the equations and name the property.
9. 4 x (7 x 2) = (4 x 7) x ____ 10. 11 + 7 = 7 + ____
11. 11 x ____ = 11 12. 3 x 2 x 8 = 8 x 3 x ___
13. 8 x ___ = 0 14. 9 + ___ = 9
15. 7 + (4 + 9) = ( __ + 4) + 9 16. 3 ( 4 + 1) = (__ x 4) + ( __ x 1)
Properties
Fill in each of the table with an example of each of the properties!
Properties of Addition
Property Example:
Commutative
Associative
Identity
Properties of Multiplication
Property Example:
Commutative
Associative
Identity
Zero
Distributive
Distributive Property
Distribute – to give something to each group!
( + )
Distributive Property
There are two different ways we can show hot items are distributed using mathematical
expressions. Both ways will give the same result in the end, they will be equal!
Example: If there are 4 students and each brought 2 folders and one notebook to class.
We can find how many supplies they brought two different ways.
4( 2 + 1) 4( 2 + 1)
2(3) 8 + 4
12 12
Order of Operations- Distributive Property 4( 2+ 1) = 4(2) = 4 (1)
solve parenthesis first.
Use the Distributive Property to complete the table
Expression Rewrite Expression Evaluate
1) 2 (4 + 1)
2) 7 (8 + 4)
3) 9( 3 + 9)
4) 2 ( 4 + 2 + 6)
5) 3 ( n + 5)
6) (a + 9)
7) 5 ( x + y + z)
Distributive Property
8) Each day, you run on a treadmill for 10 minutes and lift weights for 15 minutes. Write
and solve an expression using the distributive property to find out how many minutes of
exercise you do in 5 days.
Use the following information for number 9.
Prices
Museum Exhibit
Child (under 8) $8 $4
Student $12 x
9) A class of 20 students visits an art museum and a special exhibit while there. Use the
distributive property to write and simplify an expression for the cost.
Use the Distributive Property to solve each of the following.
10) 5 x 24 11) 6 x 81 12) 7 x 93
Use the Distributive Property to solve each of the following area models.
13) 2 6 14) 2 10
2 4
Distributive Property
Use the Distributive Property to complete the table
Expression Rewrite Expression Evaluate
1) 2 (x + 1)
2) 5 (b + 4)
3) 9( 3 + 9)
4) 2 ( x + y + 6)
5) 3 ( n + 5)
6) 8 (a + 9)
7) 5 ( x + z)
Use the Distributive property to solve each of the following!
8) 5 x 25 9) 2 x 42 10) 7 x 63
Translating Expressions
Expressions are mathematical phrases (sentences)
Numerical Expression Algebraic Expressions
*numbers and operation signs only
(all of your order of operation problems)
2 x (7-4) + 10
*numbers, operation signs and variables
14 + n nn-1
Key Vocabulary:
Variable- ______________________________________________________________
Coefficient- ___________________________________________________________
x + 5
x – 5
5x
5
x
Remember….
When translating an algebraic expression using variables:
DO NOT USE THE MULTIPLICARION SIGN!!
Why? Which of the following is the variable? 4 x f
We now write 4f when translating four times f. When the 4 is next to the (f) it means
multiplication!
More Than / Less Than
___________________
From
___________________
Translating Expressions
Identify the variable and coefficient in each expression below.
1) 9 – 8n 2) 3k + 5 3) 22 – p + 6r
Write each phrase as an algebraic expression.
4) 7 less than m
5) The quotient of 3 and
y
6) The total of 5 and c
7) The difference of 6
and r
8) N divided by 2
9) The product of k and
9
10) The Jets won 5 more games than they lost. Let “L” represent the amount of games the Jets
lost. Write an algebraic expression to show the number of games they won.
11) Lucas has 2 times as much money as Hannah. Let “m” represent the amount of money
Hannah has. Write an algebraic expression to show the amount of money that Lucas has.
12) Kara is 4 years older than Donna. Let “a” represent Donna’s age. Write an algebraic
expression to show ow old Kara is.
Evaluating Expressions
2 Types of Expressions
Numerical Expression Algebraic Expression
Numbers and operations only!
Ex: 16 + 12 ÷ 4
Operations, numbers and variables!
Ex: n+ 12 ÷ 4
Steps to Evaluating Expressions!
1) Write the original algebraic expression
2) Substitute (replace) the variable with the correct number.
3) Use order of operations to solve.
Substitution = replace the variable with the value given!
Evaluate each of the following when n = 3 and x = 4.
1) n + 4
2) 6n
3) nx
4) 9
n
Evaluate each of the following when Evaluate if a = 2 and b = 3
5) ab 6) 10 – 1 7) 9a + b
Combining Like Terms
“Like Terms” can be thought of as a family of people!
Families always come together!
Vocabulary
Variable- a symbol that represents an unknown number
Coefficient- the number in front of the variables(s) in a term
Term- when addition or subtraction signs separate an algebraic expression in to parts, each part
is a term.
Like Terms- terms that have the same variable raised to the same power but, can have different
coefficients
Constant- number all by itself
Combine- put something together (SIMPLIFY!)
Identify the variables, coefficients, constants, terms and like terms.
1) 9x + 3 – 4x 2) 3b + 5n + 2p – 11
Variables- Variables-
Coefficients- Coefficients-
Constants- Constants-
Terms- Terms-
Like Terms- Like Terms
Combining Like Terms
Determine whether each of the following are like or unlike terms.
1) Apple, Orange
2) Broccoli, Asparagus
3) 10x, 9
4) 12x, 12y
5) 10x, 2x
6) 2 apples, 3 apples
7) 8a, 2a
8) 19c, 23c
9) 25, 98f
10) 3 fire engines, 9 fire engines
11) 5t, 9t
12) 34x, 27x, 12x
13) 5 quarters, 9 dimes
14) 12w, 25w
15) W, a, t, r
16) 2a, 5a, 7a
Rules to Combine Like Terms!
1) Use shape (or color) to determine like term families
2) Add the coefficients of the like terms together
3) Bring the variables along
Examples:
1) 13b + 2b 2) 8a + 5b 3) 9m – 4m
4) 2x + 4 + 3x 5) 8a + 5a + 9 -1 6) 3x – 2x + 2y
Combining Like Terms
Try these on your own: Simplify the expressions
1) 5x + 3x 2) 8y + 4y 3) 6w + 3w + w
4) 2x2+ x2+ 8 5) 3x + 2x + y + 3y 6) 7 + 6x + 2 + 3x
7) x + 2x + 2y + 3y + 8 9) 9x + 8y + 4 + 2x + y – 2 10) 0.5x – 0.25x – 2
11) 7y – 0.2x + 0.4x – 0.9y 12) 8 + 0.7y – 8 13) 0.23z – 10 + 0.32
14) 1
2𝑥 −
1
3𝑥 − 8 15)
1
4−
1
6𝑧 + 5 +
9
2𝑥 16)
3
4𝑑 +
2
3𝑥 −
1
8𝑑 +
2
6𝑥
17) 51
2𝑥 − 10 + 9
5
8 𝑦 18) 10
1
2𝑤 − 9
1
3𝑘 − 8𝑤 19) 5k – 0.34g + 1g – 7
Challenge: 3x + 4y + 8w - 20 + 3ab + 8x + 2ab + 4y + 5w – 10
Distributing Expressions
Distributive Property- is used to multiply a single term and two or more terms inside a set
of parentheses.
Is 6(13) and 6(10 + 3) the same?
Equivalent expressions- expressions that have the same value.
The expressions 3( 2 + 4) and (3 x 2) + ( 3 x 4) are equivalent expressions because they have
the same value, 18
What does it mean to distribute?
Example: We want to distribute cookies to both Frosty and Santa so, that each of them will
get cookies!
( + ) ( ) + ( )
Distributing Expressions
Use the distributive property to write an equivalent expression. Then, use it to evaluate the
expression.
1) 2( 3 + 9)
2) 2( 6 + 4)
3) 5(5 + 3)
4) 7(9 + 11)
Rewrite using the distributive property in terms of x. (With variables)
5) 5 (2x + 5)
6) 3 ( 2 + 10b)
7) 7 ( 2x – 6)
8) 9 (4w + 12)
9) 8 ( 8x + 9)
10) 3(5t + 9 )
11) 4 ( 7t + 8 + z)
12) 2( 8y + 9x – 1)
13) 6(x + y + 5)
14) 8(4x + 2x – 2)
Factoring Numerical Expressions
Vocabulary
Greatest Common Factor- (GCF) The greatest common factor is the largest common factor of
two or more given numbers
Numerical Expressions- a number phrase without an equal sign (numbers and operation signs
only)
Equivalent Expressions- two or more expressions that have the same value.
**You can use the greatest common factor (GCF) and the Distributive Property to factor
numerical expressions.
Factoring- reverse Distributive Property
Example 1: 27 + 39
Step 1: Find the GCF of 27 and 39.
Step 2: Divide 27 and 39 by the GCF.
Step 3: Rewrite the expression as the Distributive Property
Example 2: Find an equivalent expression for 15 + 45
Factoring Numerical Expressions
Practice:
1) 24 + 40 2) 18 + 30 3) 21 + 42
4) 27 + 54 5) 60 + 12 6) 36 + 54
7) 35 + 45 8) 28 + 49 9) 16 – 12
10) 9 + 21 11) 80 – 56 12) 36 + 30
13) 63 + 9 + 81 14) 12 + 24 + 26 15) 4 + 66 + 88
16) 49 – 28 17) 102 – 10 + 2 18) 45 + 60
Factoring Numerical & Algebraic Expressions
Vocabulary
Greatest Common Factor- (GCF) The greatest common factor is the largest common factor of
two or more given numbers
Numerical Expressions- a number phrase without an equal sign (numbers and operation signs
only)
Algebraic Expressions- a combination of variables, numbers and at least one operation. ** NO
equal sign
Equivalent Expressions- two or more expressions that have the same value.
Example 1: Factoring Algebraic Expressions.
Factor the following algebraic expressions using the distributive property and the GCF
27x – 39
Step 1: Find the GCF
Step 2: Divide the terms by the GCF
Step 3: Bring down all the variables
Example 2:
Factor the following algebraic expressions using the distributive property and the GCF
36x + 120
Step 1: Find the GCF
Step 2: Divide the terms by the GCF
Step 3: Bring down all the variables
Factoring Numerical & Algebraic Expressions
Practice:
1. 15m + 35n
2. 40 k – 25 c
3. 6x + 16
4. 12b + 24
5. 20x + 25
6. 6y + 3
7. 7x – 56
8. 18x + 42y
9. 8x + 12y + 20
10. 48x – 32r – 8
11. 8x – 5y
12. 8c – 2
13. 3b + 27
14. 2x + 10 + 8y + 6b
15. 18x + 24b + 36a + 54
16. 20 + 35x + 15b + 40a + 90t
17. Bonus: a + 22b + 44c + 98d + 16t + 88m + 100