WARM-UP EXERCISES

Insert grouping symbols to make the sentences true.

1.) 7- 4 x 6 + 4 = 30

2.) 6 + 2 x 3 - 4 = 20

3.) 9 - 4 x 2 + 5 = 15

4.) 5 x 8 - 6 2 = 5

WARM-UP EXERCISES

Exponents & Powers

• Exponents are used to represent repeated multiplication.

• 46

4 is the base. The base is the number that is being repeated.

6 is the exponent. The exponent tells you how many times to repeat the base.

Exponents & Powers

• 46 - Read Four to the sixth power.

• 72 - Read Seven to the second power or

Seven squared

• 53 - Read Five to third power or

Five cubed

Exponents & Powers

• Use an exponent to represent the expression:

8 8 8 8 8

85

3x 3x 3x 3x

(3x)4

Exponents & Powers

• Evaluate:

34

3 3 3 3

81

55

5 5 5 5 5

3125

ORDER OF OPERATIONS

1.) Do operations inside grouping symbols.

2.) Evaluate exponents.

3.) Do multiplication or division in order from left to right.

4.) Do addition or subtraction in order from left to right.

ORDER OF OPERATIONS

3 42 + 1

3 16 + 1

48 + 1

49

23 - [ ( 45 15)3 9 ]2

23 - [ 33 9 ]2

23 - [ 27 9 ]2

23 - 32

23 - 9

14

Write in exponential form.

1.) p squared 2.) t t 3.) 5 t t t t

Evaluate each power.

4.) 64 5.) 25 6.) 93

Evaluate using the order of operations.

7.) 6 3 + 2 78.) 10 (3 + 2) + 9

9.) [(7 – 4)2 + 3] + 15

VARIABLES IN ALGEBRA

Variable - a letter that is used to represent one or more numbers.

Evaluating an Algebraic Expression:1.) Write the algebraic expression.2.) Substitute values for the variables.3.) Simplify the numerical expression using

the order of operations.

Examples

8x + 9y when x = 8 & y = 128 8 + 9 1264 + 1081725x + (3x + 8) when x = 45 4 + (3 4 + 8)5 4 + (12 + 8)

5 4 + 2020 + 20 = 40

Assignment1.) (x + y)2

when x = 5 & y = 3

2.) m – n2

when m = 25 & n = 4

3.) (a – b)4

when a = 4 & b = 2

4.) (d – 3)2

when d = 16

5.) c3 + d

when c = 4 & d = 16

6.) 16 + x3

when x = 2

7.) 3 + 2x3

when x = 4

8.) 3r2 – 17

when r = 6

9.) x/7 + 16

when x = 21

10.) 6 + 2p2

when p = 5

Homework Answers

P. 12

41.) 64

42.) 9

43.) 16

44.) 80

45.) 100

46.) 24

P. 19

13.) 19 14.) 32

15.) 300 16.) 213

17.) 34 18.) 91

19.) 18 20.) 24

Real Numbers

Natural Numbers– Positive whole numbers.

0, 1, 2, 3, …

Some people do not include 0 with the set of Natural Numbers.

Integers – Positive and negative whole numbers.

… -3, -2, -1, 0, 1, 2, 3 …

Rational Numbers – A number that can be written as the quotient of two integers. A rational number either terminates or is a repeating decimal.

-2, ½, ¾, √4 and 1/3

Irrational Numbers – A number that cannot be written as the quotient of two integers. Irrational numbers cannot be written as repeating or terminating decimals.

√5, 1/7, π

Examples

Identify as a natural, integer, rational or irrational number. You may have more than one answer.

1.-22

2.5/7

3.√49

4.12

Adding Real Numbers

If the signs are alike, then add the numbers together and keep the sign of the largest number.

- 5 + (-4)

5 + 4 = 9

- 9

If the signs are different, then subtract the numbers and keep the sign of the largest number.

-12 + 36

36 - 12

24

24

Examples:-7 + (-6)

-13

2.9 + ( -3.6) + (-3)

2.9 + (-6.6)

6.6 - 2.9 = 3.7

-3.7

Use your addition rules to answer the following problems.

1.) –25 + 57

2.) –45 + (-33)

3.) 66 + (-103)

4.) –12.5 + 32.4

5.) –23.6 + (-54.7)

6.) 54 + (-23) + 45

7.) –43 + (-67) + (-89)

8.) 2.6 + (-3.4) + 5.8

Subtracting Real NumbersTo subtract real numbers, change the problem to

an addition problem and solve.

SUBTRACTION TO ADDITION: (Copy/Change/Change)

1.) Keep the first number the same.

2.) Change the subtraction sign to an addition sign.

3.) Change the number behind the subtraction sign to its opposite.

4.) Follow the addition rules.

Examples:

- 4 - 7

- 4 + (-7)

-11

13 - (-35)

13 + 35

48

8 - (-6) - 7 + 14

8 + 6 + (-7) + 14

14 + (-7) + 14

7 + 14

21

Use your subtraction rules to answer the following problems.

1.) 75 – (-25)

2.) –54 – (-23)

3.) 76 – (-130)

4.) –52.1 – 43.2

5.) –6.32 – (-74.5)

6.) 45 – (-32) – 54

7.) –34 – (-76) + (-98)

Multiplying and Dividing with positives and negatives:

1.) Multiply or divide the numbers first.

2.) Place your sign on your answer.

a.) If the signs are the same, then my answer is positive.

b.) If the signs are different, then my answer is negative.

DIVISION OF REAL NUMBERS

1. Divide the numbers. (If fractions are involved, change the division to multiplication and flip the fraction behind the division sign.)

2. Place a sign on your answer. (Same rules as multiplication.)

Examples

1. 51 (-3)

2. -216 ÷ (-12)

3. -5 2/3

Examples

4. 48

- ½

3. -3x ÷ 2/5

Combining Like Terms

Terms are separated by “+” and “-” signs.

Terms having the same variable and same exponent can be combined. Use Addition rules.

Terms that do not have a variable may be combined.

1.) 7x + 12x

19x

2.) 8y – 5y + 8

3y + 8

3.) -5x + x – 10 + 8

-5x + 1x – 10 + 8

-4x – 2

Homework Answers

8.) –12t + 30 9.) –3x + 6

10.) 11b + 12 11.) 3h – 3

12.) –10k + 10 13.) 15a – 5

14.) 9c – 5 15.) 1d – 3

16.) 3h – 7 17.) 8x + 7

18.) 4z + 5 19.) 2y + 12

20.) 8p 21.) 10m + 3

Adding and Subtracting Polynomials

STANDARD FORM OF A POLYNOMIAL: 

5x3 – 4x2 + 2x – 3

*Always written in descending order, beginning with the term with the highest exponent first and ending with the constant or the term with the smallest exponent. 

Coefficient – The numbers that come before the variables.

Steps for Adding & Subtracting

1.Get rid of parentheses. (put a “1” in front of parentheses that do not have a number and multiply each term.)

2.Combine your like terms.

3.Write in standard form.

Warm – Up Exercises

1.) 3 + 8 3 ÷ 2

2.) 3 • 23 + 4 63.) 5 (12 + 3 4)

Properties of Addition & Multiplication

• Associative Property of Addition

(a + b) + c = a + (b + c)

• Associative Property of Multiplication

(a • b) • c = a • (b • c)

• In the associative property the order does not change. What is in parentheses does.

• Commutative Property of Additiona + b = b + a

• Commutative Property of Multiplicationa • b = b • a

• In the commutative property, the order of the problem does not matter.

• Identity Property of Addition0 + a = a

• Identity Property of Multiplication1 • a = a

1. 6 • 4 = 4 • 6

2. 5 1 = 5

3. (5 + 3) + 8 = 5 + (3 + 8)

4. (12 + 10) + 22 = 22 + (12 + 10)

5. 12 + 0 = 12

Finish the problem (Use the properties):

6. 5 + 3 =

7. (x + y) + z =

8. -22 + 0 =

9. 25 1 = 10. 2(4) =

1. 6 • 4 = 4 • 6 [ CPM ]

2. 5 1 = 5 [ IPM ]

3. (5 + 3) + 8 = 5 + (3 + 8) [ APA ]

4. (12 + 10) + 22 = 22 + (12 + 10) [ CPA ]

5. 12 + 0 = 12 [ IPA ]

Finish the problem (Use the properties):

6. 5 + 3 = 3 + 5 [ CPA ]

7. (x + y) + z = x + (y + z) [ APA ]

8. -22 + 0 = -22 [IPA]

9. 25 1 = 25 [IPM]

10. 2(4) = 4(2) [CPM]

Multiplying Monomials

• When two monomials have the same variable, they can be multiplied

1. Multiply the numbers.

2. Add the exponents on the like variables.

Ex: 5y2 (-7y4)

= -35y6

Multiplying Monomials by Polynomials

• –5t ( 2t2 – 7t + 9)

-5t (2t2) -5t (-7t) -5t (9)

-10t3 + 35t2 – 45t

• 3x3( 2x3 – 3x2 + 5x – 9)

3x3(2x3) 3x3(-3x2) 3x3(5x) 3x3(-9)

6x6 – 9x5 + 15x4 -27x3

Get your homework assignment out and work these problems.

1.) x (3x – 5) + 2 (3x – 5)

2.) 3x (4x – 2) – 6 (4x – 2)

3.) 5x2 (2x + 6) – 3x (2x + 6)

Homework Answers

1.) 10t3 -16t2 + 33t 2.) -21y3 + 49y2 + 9y

3.) -4x3 + 15x2 – 2x 4.) 10b3 – 28b2 – 11b

5.) 3d3 + 4d2 – 35d

Multiplying Polynomials

1. Multiply the 2nd parentheses by the first term in the 1st parentheses.

2. Multiply the 2nd parentheses by the second term in the 1st parentheses.

3. Combine like terms.

MatricesMatrix – A rectangular arrangement

of numbers into rows and columns. 3 1 0 rows -1 2 4  columns

A matrix is identified by the number of rows and columns.

 rows x columns The matrix on the previous slide has 2 rows

and 3 columns so it is 2 x 3 matrix. Read “2 by 3”.

 The numbers in a matrix are called entries.

In the matrix on the previous slide, the entry in the second row and third column is 4.

Adding & Subtracting Matrices

You cannot add or subtract matrices that have different numbers of rows and columns.

Use your addition rules and add the corresponding entries and put them in the same spot in your matrix answer.

Subtracting Matrices

1.) Change to addition. (First matrix stays the same, subtraction sign becomes addition, EACH ENTRY IN THE SECOND MATRIX MUST BE CHANGED TO ITS OPPOSITE.

 2.) Use your rule for adding matrices to get

your answer.