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MM150 Unit 1 Seminar Agenda Welcome and Syllabus Review A Review of sets of Numbers Sections

3 Kimberly Bracey AIM: kimberlyabracey Office Hours by Appointment 3

4 Syllabus Under Course Home: Syllabus and in Doc Sharing Attendance requirements Due dates Late policies Plagiarism 4

5 Seminar Show up on time Participate often Participate in a respectful manner Stay on topic Stay until the end Archived, so you can go back and review Have 2 choices, you only have to attend once. Wednesday, 10:00 PM ET, or Friday, 11:00 AM ET 5

6 Discussion Respond to all discussion questions Respond to at least 2 classmates for each discussion question. Say more than “Nice work.” 6

7 Symbols Multiplication * (shift + 8) or () or [] Square root sqrt[16] = 4 Division / 7

Sets of Numbers Natural Numbers: {1, 2, 3, 4, …} Whole Numbers: {0, 1, 2,3, …} Integers: {…-3, -2, -1, 0, 1, 2, 3, …} Rational Numbers: ½, 0.5, -6, Irrational Numbers:, √[2], √[3] Real Numbers: all rational and irrational numbers

9 Example: Find all factors of 12 An easy way to approach this task is to think of pairs of factors you could use, then make the final list from them. 1*12 AND 2*6 AND 3*4 Make sure you have every factor pair listed! Therefore, the factors of 12 (in numerical order) are 1, 2, 3, 4, 6, and 12. 9

10 EVERYONE: Find all factors of * 56 2 * 28 4 * 14 7 * 8 Therefore, the factors of 56 (in numerical order) are 1, 2, 4, 7, 8, 14, 28, and

11 Example: Determine the GCF of 12 and 56. We have already created these lists, so I will just put them under each other here: 12: 1, 2, 3, 4, 6, 12 56: 1, 2, 4, 7, 8, 14, 28, 56 Now, just plain old COMMON FACTORS of 12 and 56 include 1, 2, and 4. The GCF is 4. 11

12 Factor Tree = 2 * 2 * 2 * 3 * 3 72 = 2 3 * 3 2

13 Steps to Finding GCF There are two steps: Write down only the COMMON PRIME FACTORS (the big numbers; save the exponents for the next step). (For only the common prime factors) given the choice of powers, use the LOWEST POWER for each prime factor. 13

14 GCF Example Using Prime Factorization Find GCF (72, 150). 72 = 2 3 * = 2 * 3 * 5 2 GCF(72, 150) = 2? * 3? GCF(72, 150) = 2 * 3 = 6 14

15 Steps to Finding the Least Common Multiple There are two steps: Write down the PRIME FACTORS with the greatest exponent. Determine the product of the prime factors. 15

16 LCM example using Prime Factorization LCM(72, 150) 72 = 2 3 * = 2 * 3 * 5 2 LCM(72, 150) = 2 ? * 3 ? * 5 ? LCM(72, 150) = 2 3 * 3 2 * 5 2 =

17 Addition of Integers Same sign = = (-8) = (-5) = -7 Opposite sign 3 + (-4) = -1 Think: 4 – 3 = 1. Then take sign of larger, = 2 Think: 9 – 7 = 2. Then take sign of larger, 2. 17

Subtraction of Integers Example 1: 4 – 9 (positive four minus positive nine) = 4 + (-9) (positive four plus negative nine) = -5 (by the different signs rule of addition) Example 2: -3 – 7 (negative three minus positive seven) = -3 + (-7) (negative three plus negative seven) = -10 (by the same sign rule of addition)

Subtraction of Integers Example 3: -12 – (-14) (negative twelve minus negative fourteen) = (negative twelve plus positive fourteen) = 2 (by the different signs rule of addition)

20 Multiplication and Division of Integers Two positives = positive Two negatives = positive One of each sign = negative Examples: (3)(-2) = -6 -9/ (-3) = 3 20

21 Simplifying Fractions 15/45 Divide both the numerator and denominator by /45 = (15 / 15) / (45 / 15) = 1/3 21

22 Mixed Numbers 2 7/8 Write 2 7/8 as an improper fraction. 2 7/8 = 2 + 7/8 2/1 + 7/8 16/8 + 7/8 23/8 2 7/8 = (2*8 + 7) / 8 = 23/8 22

23 Converting an Improper Fraction to a Mixed Number 58/4 14 2/4 14 1/ /

Multiplying Fractions and Mixed Numbers 3 ½ * 1 ¼ = 7 * = 35 8 ½ * 3/4 = 1 * = 3 8

Dividing Fractions 1 ÷ *

Dividing Mixed Numbers 9 ½ ÷ 4 3/5 = 19 ÷ = 19 * = 95 46

Radical Expressions The radical symbol looks like this: √x and the x that is located within or under the radical is called the radicand. An expression that contains a radical is called a radical expression. The following is the square root of a: 2 √a and this is also a radical expression. The small 2 in front of the radical is known as the index and it indicates that this is a square root. When no index is present, then the radical is understood to be a square root with an index of 2.

28 Here are the perfect squares: (the right side of the equal sign) 0 2 = = = = = = = = = = = = = = = = = = = = 361

29 √32 = √(16 * 2) = √16 * √2 = 4 √2 √50 = √(25 * 2) = √25 * √2 = 5 √2

30 Adding and Subtracting Radical Terms Radicals are “things”… Example: 2√5 + 4√5 = 6√5 2 apples + 4 apples = 6 apples Example: 2√3 + 4√5 = 2√3 + 4√5 (can’t combine) 2 oranges + 4 apples = 2 oranges + 4 apples 30

31 Simplify: 8 sqrt[11] + 2 sqrt[11] (8 + 2) sqrt[11] 10 sqrt[11] Simplify: 13 sqrt[2] + 8 sqrt[2] (13 + 8) sqrt[2] 21 sqrt[2]

32 Multiplying Radicals sqrt[25] * sqrt[4] sqrt[100] 10 NOTE: sqrt[25] = 5 and sqrt[4] = 2 sqrt[25] * sqrt[4] 5 * 2 10 Either way you get the same answer 32

33 Dividing Radicals sqrt[36/9] sqrt[36]/sqrt[9] 6/3 2 NOTE: 36/9 = 4 sqrt[36/9] sqrt[4] 2 Either way you end up with same answer

34 Exponents So, if you’re presented with 2*2*2*2*2*2*2, you can rewrite this as 2^7 or 2 7. Beware of this situation: -2 4 vs. (-2) = -(2)(2)(2)(2) = -16 (-2) 4 = (-2)(-2)(-2)(-2) = 16 34

35 PRODUCT RULE OF EXPONENTS. (a x ) * (a y ) = a (x + y) (KEEP THE BASE and ADD THE EXPONENTS.) 2 3 * 2 2 = 2 (3 + 2) = * 5 8 = 5 (7+8) = 5 15 QUOTIENT RULE OF EXPONENTS. (a x ) / (a y ) = a (x - y) (KEEP THE BASE & SBTRCT THE EXPONENTS) 5 7 = 5 (7-5) = 5 2 = = 1 = (14-3)

36 POWER RULE OF EXPONENTS. (a x ) y = a xy ( Keep The Base and MULTIPLY THE EXPONENTS.) (2 2 ) 3 = 2 (2*3) = 2 6 = 64 (8 11 ) 4 = 8 (11*4) = 8 44 Anything to the zero power is 1. a 0 =1, a ≠ = 1; (-10) 0 = 1; 23 0 = 1; = 1 Anything to the first power is itself. a 1 =a 8 1 = 8; (-1/2) 1 = -1/2; 25 1 = 25 36

37 A negative exponent moves the term to the other side of the fraction bar. a -1 = 1/a and 1/a -1 = a 6 (-3) = 1/6^319 (-4) = 1/19^4

38 Scientific Notation 3.1 x x x

39 Converting from Scientific Notation to Decimal Notation x = Another way to look at it: x = x 1/100,000 = /100,000 =

40 Converting from Scientific Notation to Decimal Notation 1.89 * ,890 Another way to look at it 1.89 * * 1,000 1,890 40

41 Converting from Decimal Notation to Scientific Notation Convert 45,678 to scientific notation x 10 4 Convert to scientific notation 8.2 x