Just-In-TimeReview Sections 7-9 JIT7: IntegersasExpo- nents - - PowerPoint PPT Presentation

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Just-In-TimeReview Sections 7-9 JIT7: IntegersasExpo- nents - - PowerPoint PPT Presentation

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Just-In-TimeReview Sections 7-9 JIT7: IntegersasExpo- nents Natural Number Exponents If n is a natural number, x n = x x x . . . x n times Natural Number Exponents If n is a natural number, x n = x x x . . . x n times When we use

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SLIDE 1

Just-In-TimeReview

Sections 7-9

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SLIDE 2

JIT7: IntegersasExpo- nents

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SLIDE 3

Natural Number Exponents

If n is a natural number, xn = x · x · x . . . x n times

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SLIDE 4

Natural Number Exponents

If n is a natural number, xn = x · x · x . . . x n times When we use this notation, x is called the base and n is called the exponent.

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SLIDE 5

Natural Number Exponents

If n is a natural number, xn = x · x · x . . . x n times When we use this notation, x is called the base and n is called the exponent. Examples:

  • 1. 92 = 9 · 9 = 81
  • 2. 23 = 2 · 2 · 2 = 4 · 2 = 8
  • 3. (a + 2)5 = (a + 2)(a + 2)(a + 2)(a + 2)(a + 2)
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SLIDE 6

Integer Exponents

If m is an integer, x−m =

1 xm (and also 1 x−m = xm).

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SLIDE 7

Integer Exponents

If m is an integer, x−m =

1 xm (and also 1 x−m = xm).

Examples:

  • 1. 2−4 = 1

24 = 1 2 · 2 · 2 · 2 = 1 16 2. 1 5−3 = 53 = 125

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SLIDE 8

Properties of Exponents

  • 1. xa · xb = xa+b

24 · 22 = 2 · 2 · 2 · 2 · 2 · 2 = 26

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SLIDE 9

Properties of Exponents

  • 1. xa · xb = xa+b

24 · 22 = 2 · 2 · 2 · 2 · 2 · 2 = 26

  • 2. xa

xb = xa−b 45 43 = 4 · 4 · 4 · 4 · 4 4 · 4 · 4 = 42

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SLIDE 10

Properties of Exponents

  • 1. xa · xb = xa+b

24 · 22 = 2 · 2 · 2 · 2 · 2 · 2 = 26

  • 2. xa

xb = xa−b 45 43 = 4 · 4 · 4 · 4 · 4 4 · 4 · 4 = 42

  • 3. (xa)b = xab

(54)3 = 54 · 54 · 54 = 54+4+4 = 512

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SLIDE 11

Properties of Exponents

  • 1. xa · xb = xa+b

24 · 22 = 2 · 2 · 2 · 2 · 2 · 2 = 26

  • 2. xa

xb = xa−b 45 43 = 4 · 4 · 4 · 4 · 4 4 · 4 · 4 = 42

  • 3. (xa)b = xab

(54)3 = 54 · 54 · 54 = 54+4+4 = 512

  • 4. (xy)a = xaya

(2z)3 = 2z · 2z · 2z = 2 · 2 · 2 · z · z · z = 23z3

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SLIDE 12

5. x y a = xa ya 7 2 4 = 7 2 · 7 2 · 7 2 · 7 2 = 7 · 7 · 7 · 7 2 · 2 · 2 · 2 = 74 24

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SLIDE 13

5. x y a = xa ya 7 2 4 = 7 2 · 7 2 · 7 2 · 7 2 = 7 · 7 · 7 · 7 2 · 2 · 2 · 2 = 74 24 6. x y −a = y x a 4 3 −2 = 4−2 3−2 =

1 42 1 32

= 1 42 · 32 1 = 32 42

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SLIDE 14

5. x y a = xa ya 7 2 4 = 7 2 · 7 2 · 7 2 · 7 2 = 7 · 7 · 7 · 7 2 · 2 · 2 · 2 = 74 24 6. x y −a = y x a 4 3 −2 = 4−2 3−2 =

1 42 1 32

= 1 42 · 32 1 = 32 42

  • 7. x−a

y−b = yb xa 5−3 2−4 =

1 53 1 24

= 1 53 · 24 1 = 24 53

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SLIDE 15

Special Cases

x1 = x x0 = 1 as long as x = 0. (00 is undefined.)

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SLIDE 16

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4

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SLIDE 17

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3

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SLIDE 18

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10

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SLIDE 19

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10 y6 3x

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SLIDE 20

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10 y6 3x 3. 2a2 b 4 b2 4a3 2

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SLIDE 21

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10 y6 3x 3. 2a2 b 4 b2 4a3 2 a2

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SLIDE 22

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10 y6 3x 3. 2a2 b 4 b2 4a3 2 a2 4. 2q−4r−3s 3r4s−3 −2

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SLIDE 23

Examples

Simplify the following expressions and eliminate any negative exponents.

  • 1. (3x)3y

xy4 27x2 y3 2. 3x2y−4 9x3y−10 y6 3x 3. 2a2 b 4 b2 4a3 2 a2 4. 2q−4r−3s 3r4s−3 −2 9r14q8 4s8

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SLIDE 24

JIT8: ScientificNotation

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SLIDE 25

Definition of Scientific Notation

A number is in scientific notation if it is written in the form a × 10n where a has exactly one non-zero digit left of the decimal point, and n is an integer. For example: 2.37 × 106 and −1.0021 × 10−100.

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SLIDE 26

Examples

Convert the following numbers from scientific notation to decimal notation:

  • 1. −3.001 × 105
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SLIDE 27

Examples

Convert the following numbers from scientific notation to decimal notation:

  • 1. −3.001 × 105
  • 300,100
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SLIDE 28

Examples

Convert the following numbers from scientific notation to decimal notation:

  • 1. −3.001 × 105
  • 300,100
  • 2. 2.42 × 10−3
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SLIDE 29

Examples

Convert the following numbers from scientific notation to decimal notation:

  • 1. −3.001 × 105
  • 300,100
  • 2. 2.42 × 10−3

0.00242

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SLIDE 30

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000
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SLIDE 31

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000

2.504 × 107

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SLIDE 32

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000

2.504 × 107

  • 2. 1.3
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SLIDE 33

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000

2.504 × 107

  • 2. 1.3

1.3 × 100

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SLIDE 34

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000

2.504 × 107

  • 2. 1.3

1.3 × 100

  • 3. −0.09624
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SLIDE 35

Examples

Convert the following numbers from decimal notation to scientific notation:

  • 1. 25, 040, 000

2.504 × 107

  • 2. 1.3

1.3 × 100

  • 3. −0.09624

−9.624 × 10−2

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SLIDE 36

JIT9: OrderofOperations

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SLIDE 37

Rules

  • 1. Simplify inside grouping symbols first (for example: parenthesis,

brackets, inside square roots, top and bottom of fractions).

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SLIDE 38

Rules

  • 1. Simplify inside grouping symbols first (for example: parenthesis,

brackets, inside square roots, top and bottom of fractions).

  • 2. Evaluate exponents.
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SLIDE 39

Rules

  • 1. Simplify inside grouping symbols first (for example: parenthesis,

brackets, inside square roots, top and bottom of fractions).

  • 2. Evaluate exponents.
  • 3. Perform all multiplications and divisions, left to right.
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SLIDE 40

Rules

  • 1. Simplify inside grouping symbols first (for example: parenthesis,

brackets, inside square roots, top and bottom of fractions).

  • 2. Evaluate exponents.
  • 3. Perform all multiplications and divisions, left to right.
  • 4. Perform all addition and subtraction, left to right.
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SLIDE 41

Rules

  • 1. Simplify inside grouping symbols first (for example: parenthesis,

brackets, inside square roots, top and bottom of fractions).

  • 2. Evaluate exponents.
  • 3. Perform all multiplications and divisions, left to right.
  • 4. Perform all addition and subtraction, left to right.

The acronym PEMDAS is a common device to help remember the order: parenthesis, exponents, multiplication/division, addition/subtraction.

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SLIDE 42

Examples

Simplify each of the following:

  • 1. 3 · 15 − 4 · 24 + 2(1 − 7)

−31

  • 2. 42 · 81 ÷ 33 · 2−3

6

  • 3. 62 − 2(3 − 5)2

43 − 32 7 8