 # WARM-UP EXERCISES Insert grouping symbols to make the sentences true.

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WARM-UP EXERCISES Insert grouping symbols to make the sentences true.

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 - 9 14

Write in exponential form. 1. ) p squared 2. ) t  t 3
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   7 8.) 10  (3 + 2) ) [(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 20 + 20 = 40 8x + 9y when x = 8 & y = 12 8  8 + 9  12
172 5x + (3x + 8) when x = 4 5  4 + (3  4 + 8) 5  4 + (12 + 8) 5  = 40

Assignment 1.) (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/ when x = ) 6 + 2p2 when p = 5

Homework Answers P. 12 41.) 64 42.) 9 43.) 16 44.) 80 45.) 100 46.) 24
13.) ) 32 15.) ) 213 17.) ) 91 19.) ) 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. -22 5/7 √49 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.
24

Examples: -7 + (-6) -13 2.9 + ( -3.6) + (-3) 2.9 + (-6.6)
= 3.7 -3.7

1.) – ) –45 + (-33) 3.) 66 + (-103) 4.) – ) – (-54.7) 6.) 54 + (-23) ) –43 + (-67) + (-89) 8.) (-3.4) + 5.8

Subtracting Real Numbers
To 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 14 + (-7) + 14 7 + 14 21 - 4 + (-7) -11 13 - (-35) 48

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
Divide the numbers. (If fractions are involved, change the division to multiplication and flip the fraction behind the division sign.) Place a sign on your answer. (Same rules as multiplication.)

Examples 51  (-3) -216 ÷ (-12) -5  2/3

Examples 48 - ½ -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 – -5x + 1x – -4x – 2

Homework Answers 8.) –12t + 30 9.) –3x + 6 10.) 11b + 12 11.) 3h – 3
14.) 9c – ) 1d – 3 16.) 3h – ) 8x + 7 18.) 4z ) 2y + 12 20.) 8p 21.) 10m + 3

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.

Get rid of parentheses. (put a “1” in front of parentheses that do not have a number and multiply each term.) Combine your like terms. Write in standard form.

Warm – Up Exercises 1.)  3 ÷ 2 2.) 3 •  6 3.) 5  (  4)

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.

a + b = b + a Commutative Property of Multiplication a • b = b • a In the commutative property, the order of the problem does not matter.

0 + a = a Identity Property of Multiplication 1 • a = a

6 • 4 = 4 • 6 5  1 = 5 (5 + 3) + 8 = 5 + (3 + 8) ( ) + 22 = 22 + ( ) = 12 Finish the problem (Use the properties): = 7. (x + y) + z = = 9. 25  1 = 10. 2(4) =

6 • 4 = 4 • 6 [ CPM ] 5  1 = 5 [ IPM ] (5 + 3) + 8 = 5 + (3 + 8) [ APA ] ( ) + 22 = 22 + ( ) [ CPA ] = 12 [ IPA ] Finish the problem (Use the properties): = [ CPA ] 7. (x + y) + z = x + (y + z) [ APA ] = -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
Multiply the 2nd parentheses by the first term in the 1st parentheses. Multiply the 2nd parentheses by the second term in the 1st parentheses. Combine like terms.

Matrices Matrix – A rectangular arrangement of numbers into rows and columns. rows 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.