# Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

## Full text

(1)

### P

lease Parenthesis - Do all grouped operations first.

### E

xcuse Exponents - Second

### M

y Multiplication and Division - Left to Right.

ear

### A

unt Addition and Subtraction - Left to Right.

### S

haniqua

Follow the correct order of operations to evaluate expressions.

Evaluate: Remember to use the correct Order of Operations.

1.

2.

3.

### 

2

Evaluate for a=3, b=4, c=5, d=10

1.

2.

3.

2

### 

Solve the following using the correct order of operations:

1.

6.

2

2.

7.

3.

8.

4.

9.

5.

10.

2

(2)

### Algebra

When evaluating expressions, work using the correct order of operations: P (Parenthesis) Do all grouped operations first.

E (Exponents) Do all operations involving exponents.

M D (Mult./Div.) Do all multiplication and division from left to right. A S (Add./Sub.) Do all addition and subtraction last - from left to right. Solve: 1.

2

2.

3.

4.

5.

2 6.

2 7.

8.

2

9.

10.

2

2

### Order of Operations Practice

Name________________________ Period _____

(3)

### Algebra

Evaluate for a=3, b=4, c=6

11.

2 12.

2

2 13.

2 14.

15.

2

16.

2

17.

18.

2

19.

20.

2

### Order of Operations Practice

Name________________________ Period _____

(4)

### 1.4

notes:

Integers are positive and negative Whole Numbers like

-9 127 -90 -54 75 120 65 21 -78 -23 -11 70

Integers are NOT decimals or fractions.

Adding and subtracting integers can seem unnecessarily complicated. Try the following practice problems first:

Practice:

1.

2.

3.

4.

5.

6.

7.

8.

### )

If you got all of these right, you already have a proven method for adding and subtracting integers. Close your ears, sit quietly, and continue using your own method. If you missed even one, pay close attention and take notes.

notes:

Same Sign Sum

When adding integers with the same sign, find the sum and keep the sign of both numbers.

1.

2.

3.

4.

### )

Different Sign Difference

When adding integers with different signs, find the difference and keep the sign of the ‘bigger’ number.

1.

2.

3.

4.

1.

2.

3.

4.

5.

6.

7.

8.

(5)

### 1.4

Subtracting Integers:

SMATO Subtraction Means Add The Opposite

Subtracting Integers is more complicated than adding integers.

To subtract integers, change subtraction to addition and switch the sign of the second number. Then, follow the two rules we have learned for adding integers.

1.

2.

3.

4.

### )

Practice: Change to addition, then solve.

1.

4.

2.

5.

3.

6.

### )

Use the same rules for fractions and decimals as you would for integers: Same Sign Sum, Different Sign Difference, SMATO.

Examples: 1.

2.

3.

4.

5.

6.

Practice: 1.

2.

3.

4.

5.

6.

(6)

### Algebra

Matrices: A matrix is a rectangular table of numbers.

Horizontal lines are called rows. Vertical lines are called columns.

              9 5 3 3 2 1 A               3 2 5 7 1 0 B            1 7 3 0 1 5 1 2 C

Matrix A and B are 3x2 matrices. C is a 2x4 matrix.

Q: In matrix C, which number is in the second row, third column?

You can only add or subtract matrices with the same dimensions. Subtraction is easier if you simply add the opposite.

              9 5 3 3 2 1 A               3 2 5 7 1 0 B                12 3 8 10 1 1 B A

### A

Practice: Solve the following using the given matrices:

### C

1. A+B 2. A-C 3. B-C 4. A+C

(7)

### Algebra

Matrix multiplication: Multiplying a matrix by a scalar (a number shown outside the matrix) involves multiplying each term by the scalar.

### 2 A

                           2 2 6 0 2 8 1 1 3 0 1 4 2

Complete the following operations involving matrices:

         5 2 4 1 A        6 3 2 5 B         1 4 0 4 C 1.

2.

3.

4.

### C

Solving Matrix Equations: Ex:

### 3

One-step. Solve for B: Two-steps. Solve for A:

                   0 3 1 2 5 2 4 1 B                   1 3 5 4 7 3 7 0 2A Practice:

Solve for the missing matrix in each problem below:

1.                             1 7 4 5 2 3 9 5 3 3 2 1 A 2.                             3 7 4 1 0 3 3 9 6 1 1 5 2 A

(8)

### Algebra

Complete the following problems using the given matrices:

Write ‘impossible’ if a problem cannot be solved.

              9 5 3 3 2 1 A               3 2 5 7 1 0 B            1 7 3 0 1 5 1 2 C 1.

2.

3.

4.

5.

### B

6. Solve for matrix X:

                             3 4 1 5 6 13 7 5 1 2 4 1 X

### Matrix Operations

Name________________________ Period _____

(9)

### Algebra

Combining Like Terms:

When adding or subtracting numbers and variables, you can only combine like terms.

L i k e t e r m s contain the same variables, with the same exponents

in a single product.

Here are some sets of like terms:

2 2 2

3 3 3

### 2

Practice: Match each pair or set of like terms below:

3

2

2

3

3

2

2

2

2

2 2

3

2

### y

You cannot add or subtract unlike terms. It is like trying to add apples and oranges.

Practice: Simplify the following:

1.

2.

3.

4.

2

2 5.

6.

2

3

### ab

You cannot simplify number six because there are NO LIKE TERMS.

(10)

### Algebra

Practice: Simplify.

1.

2.

3.

4.

2

2 5.

6.

2

3

### 2.3

Practice: Simplify. 1.

2.

3. 2 2

4.

2

5. 2 2

6.

### 

Practice: Simplify. 1.

2.

3.

4.

2

2 5.

2

2

6. 2 2 2

(11)

### Like Terms Reteach:

Name________________________ Period _____

Combining like terms is just like adding and subtracting integers: Simplify.

1.

2.

3.

4.

5.

6.

7.

5

5 8.

3

3 9.

5

2

### x

5 2

You can only combine terms with the same variables and exponents: Simplify. Write SIMPLIFIED if there are no terms which can be combined. Circle like terms as you combine them.

7.

8.

9.

2

3

3

2 10.

2

2

2

2

11.

2

3

2

3 12.

2

3

3

2

13.

2

3

3

2

3 14.

2

2

2

2

(12)

### Like Terms Reteach:

Name________________________ Period _____

Fraction review: Solve.

15.

16.

17.

18.

19.

20.

### 

Simplify each by combining like terms.

21.

22.

23.

3

3

3

24. 2 2 2

25. 3 2 2 2 2

3

(13)

### Like Terms Practice

Name________________________ Period _____

Simplify each. Write simplified if no terms can be combined.

1. 2 3 2 3

2. 2 3 3 2

3.

2

3

2

3 4. 2 3 3 2

5.

2

2

2

2 6.

2 3 3 2

7.

3

3

8. 2 5 2 5

9.

2

3

2

2 10. 2 3 3 2

11. 2 3 2 3

12.

2

3

3

2

13. 2 2 2

14.

2

2

2

(14)

### Algebra

Base:

The repeated factor in a power. In the expression n³, n is the base.

Exponent:

Represents the number of times a factor is being multiplied. In the expression n³, the ³ is the exponent.

The expression

### 5

3 means that you multiply

The expression

### x

5 means that you multiply

The expression

### )

4 means that you multiply

### )

Practice: Write-out without using exponents:

1.

5 2.

4 3.

2

3 4.

3

### y

2

Practice: Write using exponents.

1.

2.

3.

### 7

Practice: Evaluate. (solve)

1.

2 2.

4 3.

3

2 4.

2

4

(15)

### Algebra

One of the easiest ways to multiply expressions using exponents is to write them out in factored form, and then recombine terms using exponents:

Ex.

2

### n

3 Practice: Simplify.

1.

2

2.

2

5

3.

23

11

12

### b

7

Rules:

When multiplying variables with exponents, simply add exponents:

Ex.

3

5

53

8 or

2

3

5

2

25

32

7

### y

5 Practice: Simplify.

1.

25

12

2.

15

30

21

14

3.

10

7

3

### n

20

The same rules apply for positive and negative exponents.

Practice: Simplify. 1.

3

4

2.

5

2

2

4

3.

3

8

2

4.

12

22 5.

13

2

11

6.

3 2 11

### y

Challenge: Find the Perimeter AND Area of each shaded figure below:

note: all angles are right angles.

(16)

### 8.5

Review: Multiply.

1.

4

2

5

2.

3

2

2 3.

3

2

### )

5 Dividing Monomials:

You can write-out variables and exponents, or simply subtract exponents:

Examples: 1.

4 3 2

2. 9 2 11 5

### x

Practice: Divide/ Simplify. Answers should have positive exponents.

1. 4 3 3

2.

9 8 12

3. 6 2 3 3

### a

 

What is a negative exponent?

Look at the following pattern in our own number system:

### 876.543

The 8 is in the ________ place

2

### .

The 7 is in the ________ place

1

### .

The 6 is in the ________ place

0

### .

The 5 is in the ________ place

1

### .

The 4 is in the ________ place

2

### .

The 3 is in the ________ place

3

(17)

### 8.5

Negative Exponents: A negative exponent can be expressed as a posi-tive exponent in the denominator:

Examples: 3 3

4 4

3 5 3 5

5 2 2 5

### 

 

Notice that a negative exponent in the denominator can also be ex-pressed as positive in the numerator.

Practice: Rewrite with positive exponents:

1.

2 2.

3 3. 9 4 

4. 2 9

5.

3 6.

### ab

2 4  

Negative Exponents: The easiest way to simplify expressions with negative exponents is to begin by rewriting them:

Examples: 3 4 2 5

  2 5

 

### x

Practice: Rewrite with positive exponents, then simplify:

1. 5 2

 

2. 9 3 2

  3. 5 2 4 3

### a

You can also use the subtraction method, but it becomes much more confusing.

(18)

### Division With Exponents

Name________________________ Period _____

1. 7

2

2. 3

2

3. 2

5

4. 5

3 3

5.

2 6. 10 15

10 30

7. 4

3

8. 5 5

3

9. 2

5

10. 7

3

11.

2 3 5

12. 2 2

3

(19)

### Division With Exponents

Name________________________ Period _____

More challenging problems:

13.

 7 12

14.

 3 15 6 20

15. 4

5

16.

 2 2 2 3

17.

 3 5 2

18.

### 

   3 5 10 10 30

(20)

### Algebra

Raising a power to a power:

Practice: Simplify each using what you know about exponents.

1.

2

4 2.

2

2 3.

12

8

### )

5

Examples: Raising a power to a power.

1.

5

2

3 2.

2

11

6 3.

## 

2 5 4 2

### y

Practice: Raising a power to a power.

1.

3

5 2.

16

21

2 3.

2 5

5 4. 5 3 3

5.

11

2 6.

2

2

3

(21)

### 8.5

Raising a fraction to a power: When raising a fraction to a power, apply the exponent to the numerator and the denominator:

Examples: 3 2

3 5 2 2

4 2 5

### y

Practice: Try these easy ones: Write-out if necessary. 1. 3

2. 5 3 2

3. 4 3 4

### x

Practice: Try these more difficult problems.

Like most of the math we have done, there are many ways to get the right answer. Answers should have positive exponents.

1. 5 5 2

2. 2 3 9 2

3. 5 2 4

  

### x

You can simplify what is in parenthesis before or after applying the exponent.

(22)

### Exponents Reteach: Multiplying

Name________________________ Period _____

Write each expression out without using exponents (write small!):

1.

3

4 2.

2

3 3.

3

3

### )

2

Rewrite each expression using exponents:

4.

5.

6.

### 5

Simplify each using the rules for exponents.

7.

12

3

4

5

8.

5

14

9.

2

7

2

10.

8

3

2 11.

5

11

5 12.

4

10

2

13.

2

3

5

2 14.

3

4

5 15.

3

2

2

2

(23)

### Exponents Reteach: Dividing

Name________________________ Period _____

Write each expression out without using exponents (write small!): ex.

3

19. 7 3

20. 3 3 2

21.

2 2

### (

Simplify each, then rewrite each expression using exponents:

22.

23.

24.

### 

Rewrite each with positive exponents: DO NOT SIMPLIFY, just rewrite using all positive exponents: 25.

4 26. 4 3 2  

27. 4 7

 

28. 2 4 2

 

### a

Rewrite each then simplify: Take your time and complete several steps.

29. 2 14 30 8

30. 5 2 3

31. 2 2 3 3

(24)

### 1.7

Cut-out the following and give each table a set (8 sets). Match the letters to the proper numbers to find the clue. ans: read it backwards (read backwards)

1.

2

2

2 s.

4 2.

2

2

2 d.

2 2 3. 2 2 2

r. 4

4.

12

3

3

3 a.

15

6 5. 6 3 3 21

w. 6 15

6. 17 11 5 2  

k. 15 6

7.

2

6

4

9

6

15 c.

6 15

(25)

8.

9

12

3

2

2 a.

15

### b

8 9. 11 15 8 15 8 11

b. 15 8

10. 15 8 15 8

t. 15 8

11. 4 4 4 11

  i. 8 15

12. 5 5 10 3

 

d. 8 15

13.

2

5

3

6

15 a.

6 15 14.

2

2

3

e.

2

2

3

15.

6

5

7

6

6

r.

5

7

(26)

### Practice Quiz: Chapter 1 (4)

Solve for a=3, b=5, c=2

1.

2

### ac

1.______ 2. b2 c(ab) 2.______ 3. (cb)(ba)2 3.______ Simplify: 4.

2

### b

2 4.__________________________ 5.

2

2

### 4

5.__________________________ 6.

4

4

### 12

6.__________________________ 7.

12

11

### x

10 7.__________________________ 8.

2

2

2

2

### b

8.__________________________ 9.

3

2

3

### a

3 9.__________________________ Name________________________ Period _____

(27)

### Practice Quiz: Chapter 1 (4)

Simplify: All answers should be written with positive exponents.

10.

5

### x

3 10.___________________ 11.

3

### ab

3 11.___________________ 12.

6

2

### )

12.___________________ 13.

2

3

### )

2 13.___________________ 14.

2

12

8

3

### )

14.___________________ 15. 9

3

### y

15.______________ 16. 5

3

### a

16.______________ 17.

### x

7 2 5 17.______________ 18.

2 2 2

### ab

18.______________ Name________________________ Period _____

(28)

### Practice Quiz: Chapter 1 (5/7)

Solve for a=-3, b=5, c=2

1.

2

### ac

1.______ 2. b2 c(ab) 2.______ 3. (cb)(ca)2 3.______ Simplify: 4.

2

### b

2 4.__________________________ 5.

2

2

### 4

5.__________________________ 6.

4

4

### 12

6.__________________________ 7.

12

11

### x

10 7.__________________________ 8.

2

2

2

2

### b

8.__________________________ 9.

3

2

3

### a

3 9.__________________________ Name________________________ Period _____

(29)

### Practice Quiz: Chapter 1 (5/7)

Simplify: All answers should be written with positive exponents.

10.

2 5

### 2x

10.___________________ 11.

3

### ab

3 11.___________________ 12.

6

2

### )

12.___________________ 13.

2

12

8

3

### )

2 13.___________________ 14.

2 2 3 2 2

### x

14.___________________ 15. 9

3

### y

15.______________ 16. 3

8

### a

16.______________ 17.

  2 7 7 2 3

### a

17.______________ 18.

2 2 2

### ab

18.______________ Name________________________ Period _____

(30)

### 1.7

The Distributive Property states:

For any numbers a, b, and c:

### 

Examples:

distribute the 5 distribute the 3

2 2

2 3 2

### x

Multiply the term outside the parenthesis by both terms inside.

Practice: Rewrite using the Distributive Property.

1.

4.

2

2.

5.

3.

2

6.

### )

Distributing the negative: Ex:

distribute the -5 distribute the -a

2

### 

distribute the negative (-1).

### y

Practice: Rewrite using the Distributive Property.

1.

4.

2

2.

5.

(31)

### 1.7

Practice: Rewrite the Following Using the Distributive Property:

1.

2.

5

2

3.

2

4.

2

5.

2

2

6.

### a

2

Practice: Fill-in the blanks. The GCF has been factored out for you.

1.

2

2

2.

2

3

2

2

3.

2

### ___)

Practice: Factor the Following (Reverse the Distributive Property)

1.

3

2 3.

2

(32)

### 1.7+

You can use the Distributive Property with division. Example: 1.

Divide

and

Therefore

Practice: 1.

2.

3.

2

4.

2

5.

5 2

6.

3 2

1.

2.

5 2

3.

4

### 

Challenge Set: Some answers will include fractions.

1.

4 3

2.

3 2

3. 5 2 2

 

(33)

### Distributive Property

Name________________________ Period _____

### 1.7+

Rewrite and Simplify using the Distributive Property:

1.

2.

3.

4.

2

5.

2

6.

2

7.

8.

9.

10.

11.

12.

3

3

(34)

### Distributive Property

Name________________________ Period _____

Simplify: Distributing division (‘bunny ears’):

13.

14.

15. 2 3 5

16.

3 2

17.

18. 5 5 6 7

19.

3 2

20. 2 2 3 5

### 

Factor (Rewrite using the Distributive Property in reverse):

21.

2

3

2

22.

2

2

3

2

(35)

### Distributive Reteach: Multiplying

Name________________________ Period _____

Rewrite each problem below using the Distributive Property.

Multiply each term outside of the parenthesis by all terms inside the parenthesis. Careful with your signs and remember your rules for multiplying with exponents.

1.

2.

2

3.

4.

5.

2

6.

2

3

2

7.

3

2

8.

3

2

2

9.

5

2

2

3

### )

Now try distributing some negatives. Remember your integer rules.

10.

11.

2

12.

2

### )

For #12 above: DISTRIBUTE THE -2 NOT THE 4a2. Answer: 4a2-2a+6.

Try the following similar problems and combine like terms wherever possible to simplify your answer.

13.

3

2

14.

5

2

5

3

15.

(36)

### Distributive Reteach: Dividing

Name________________________ Period _____

Rewrite each problem below using the Distributive Property.

‘Bunny-Ear’ each term in the numerator with the term in the denominator. Careful with your signs!

18.

19.

3

20.

3 2

21.

2 2

22.

2 3 2

23.

3 2 2

24. 1 2

25. 2 2 2 3 3 

### x

26. 13 6 24 21 11 30

### 

The final three answers involve fractions. The fractions should be simplified and left as coefficients. 27.

28.

3 2

29. 2 2 3

(37)

### Review

Name________________________ Period _____

### 1.7+

Rewrite and Simplify using the Distributive Property:

1.

2.

2

3.

2

4.

5.

2

2

6.

7.

8.

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