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Welcome to MM150!

Unit 1 Seminar

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MM150 Unit 1 Seminar Agenda

• Welcome and Syllabus Review

• A Review of sets of Numbers

• Sections 1.1 - 1.6

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Kimberly Bracey

• e-mail: Kbracey@kaplan.edu• AIM: kimberlyabracey

• Office Hours by Appointment

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Syllabus

• Under Course Home: Syllabus and in Doc Sharing

• Attendance requirements• Due dates• Late policies• Plagiarism

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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

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Discussion

• Respond to all discussion questions• Respond to at least 2 classmates for each

discussion question.• Say more than “Nice work.”

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Symbols

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

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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

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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.

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EVERYONE: Find all factors of 56.

1 * 56

2 * 28

4 * 14

7 * 8

Therefore, the factors of 56 (in numerical order) are 1, 2, 4, 7, 8, 14, 28, and 56.

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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.

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Factor Tree72

2 36

2 18

2 9

3 3 72 = 2 * 2 * 2 * 3 * 3

72 = 23 * 32

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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.

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GCF Example UsingPrime Factorization

Find GCF (72, 150).

72 = 23 * 32

150 = 2 * 3 * 52

GCF(72, 150) = 2? * 3?

GCF(72, 150) = 2 * 3 = 6

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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.

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LCM example usingPrime Factorization

• LCM(72, 150)

• 72 = 23 * 32

• 150 = 2 * 3 * 52

• LCM(72, 150) = 2? * 3? * 5? • LCM(72, 150) = 23 * 32 * 52 = 1800

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Addition of Integers

Same sign

• 4 + 6 = 10

• 12 + 3 = 15

• -3 + (-8) = -11

• -2 + (-5) = -7

Opposite sign

• 3 + (-4) = -1 Think: 4 – 3 = 1. Then take sign of larger, -1

• -7 + 9 = 2 Think: 9 – 7 = 2. Then take sign of larger, 2.

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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)

= -12 + 14 (negative twelve plus positive fourteen)

= 2 (by the different signs rule of addition)

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Multiplication and Division of Integers

• Two positives = positive

• Two negatives = positive

• One of each sign = negative

Examples:

(3)(-2) = -6

-9/ (-3) = 3

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Simplifying Fractions

• 15/45

• Divide both the numerator and denominator by 15.

• 15/45 = (15 / 15) / (45 / 15) = 1/3

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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

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Converting an Improper Fraction to a Mixed Number

• 58/4

• 14 2/4

• 14 1/223

144 / 58 4 18 16 2

Multiplying Fractions and Mixed Numbers

3 ½ * 1 ¼

= 7 * 5

2 4

= 35

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½ * 3/4

= 1 * 3

2 4

= 3

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Dividing Fractions

1 ÷ 2 3 7

1 * 73 2

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Dividing Mixed Numbers

9 ½ ÷ 4 3/5

= 19 ÷ 23 2 5

= 19 * 5 2 23

= 95 46

Radical ExpressionsThe 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.

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Here are the perfect squares:(the right side of the equal

sign) 02 = 012 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

162 = 256

172 = 289

182 = 324

192 = 361

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√32

= √(16 * 2)

= √16 * √2

= 4 √2

√50= √(25 * 2)

= √25 * √2

= 5 √2

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Adding and Subtracting Radical Terms

• Radicals are “things”… Example:

2√5 + 4√5 = 6√52 apples + 4 apples = 6 apples

Example: 2√3 + 4√5 = 2√3 + 4√5 (can’t

combine)2 oranges + 4 apples = 2 oranges + 4

apples30

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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]

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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

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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

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Exponents

• 672

• 34

• -73

• 59

• So, if you’re presented with 2*2*2*2*2*2*2, you can rewrite this as 2^7 or 27.

• Beware of this situation:

• -24 vs. (-2)4

-24 = -(2)(2)(2)(2) = -16(-2)4 = (-2)(-2)(-2)(-2) = 16

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PRODUCT RULE OF EXPONENTS.

(ax) * (ay) = a(x + y) (KEEP THE BASE and ADD THE EXPONENTS.)

23 * 22 = 2 (3 + 2) = 25

57 * 58 = 5 (7+8) = 515

QUOTIENT RULE OF EXPONENTS.

(ax) / (ay) = a(x - y) (KEEP THE BASE & SBTRCT THE EXPONENTS)

57 = 5 (7-5) = 52 = 25 93 = 1 = 1

55 914 9 (14-3) 911

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POWER RULE OF EXPONENTS.

(ax)y = axy (Keep The Base and MULTIPLY THE EXPONENTS.)

(22)3 = 2 (2*3) = 26 = 64(811)4 = 8 (11*4) = 844

Anything to the zero power is 1. a0=1, a ≠ 0

40 = 1; (-10)0 = 1; 230 = 1; 1000 = 1

Anything to the first power is itself. a1=a

81 = 8; (-1/2)1 = -1/2; 251 = 2536

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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) = 1/6^3 19(-4) = 1/19^4

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Scientific Notation

3.1 x 104

9.2346 x 10-5

1.89 x 100

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Converting from Scientific Notation to Decimal Notation

9.2346 x 10-5 = 0.000092346

Another way to look at it:

9.2346 x 10-5

= 9.2346 x 1/100,000 = 9.2346/100,000

= 0.000092346

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Converting from Scientific Notation to Decimal Notation

• 1.89 * 103

• 1,890

• Another way to look at it• 1.89 * 103

• 1.89 * 1,000• 1,890

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Converting from Decimal Notation to Scientific Notation

Convert 45,678 to scientific notation

4.5678 x 104

Convert 0.0000082 to scientific notation

8.2 x 10-6

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