8G Slides Ch 4 Exponents.ppt

(1)

8th Grade Slides

(2)

4.1.1 - Exponents

(3)

Quick Review

x

y

Base

(4)

Exponent Review

• Repeated multiplication:

5

3

=

5

·

5

·

5

(5)

Exponent Review

(6)

Exponent Review

• Any number to the first power is itself

– So

– And

4

1

=

4

(7)

Exponent Review

(8)

Exponent Review

• Any non-zero number to the power of 0 is 1.

– So,

– and (assuming x does not = 0),

50 =1

(9)

Question

• What is

(10)

Question

• What is

• Is it the same as

-

3

2

-3

(11)

Odd and Even Exponents

(12)

Odd and Even Exponents

• How do we know whether a negative number raised to a particular power will be positive or negative?

-

5 ( )

37

-

5

(13)

Odd and Even Exponents

• How do we know whether a negative number raised to a particular power will be positive or negative?

-

5 ( )

37

-

5 ( )

38

Negative base with an odd exponent results in a

NEGATIVE answer

Negative base with an even exponent results in a

(14)

Decimals and Fractions

• Is this the same as

(15)

Decimals and Fractions

• Is this the same as

No!

• How about

(16)

Negative Exponents

• Look at this progression of numbers:

33 =

32 =

31 =

30 =

3-1 =

3-2 =

(17)

Negative Exponents

• Look at this progression of numbers:

33 = 27

32 = 9

31 = 3

30 = 1

3-1 = 1/3

3-2 = 1/9

(18)

Negative Exponents

• Rule:

(19)

Negative Exponents

• The reverse is also true:

1 a

-b

=

a

(20)

Negative Exponents

• We generally do NOT want negative

exponents in our answer (unless we are using Scientific Notation).

• We move bases with negative exponents to the denominator where they will turn

positive.

– If a negative exponent is found in a

(21)

Method

Step 1: Take the RECIPROCAL of the base.

(22)

Examples

8

-3

=

x

-4

=

1

a-3 =

-4

( )

-6

(23)

Examples

8

-3

=

1

8

3

x

-4

=

1 x

4

1

a-3 = a 3

-4

( )

-6

= 1

(24)

Exception

• Numbers generally get LARGER when we square them. What kind of

(25)

Exception

• Numbers generally get LARGER when we square them. What kind of numbers get SMALLER when we square them?

• Numbers between 0 and 1 (fractions and decimals)

(26)

Classwork

(27)

4.1.3 - Exponent Laws

(28)

Exponents

• Reminders:

– Anything raised to the power of 1 is itself. – Any non-zero number raised to the power

of 0 is 1.

– When we see we use PEMDAS to determine what to do first.

• In this case, raise ‘x’ to the 3rd power, THEN multiply by 4

(29)

Product Law of

Exponents

• How would we multiply:

(30)

Product Law of

Exponents

• We could rewrite it as:

(4 x 4 x 4) x (4 x 4)

(31)

Product Law of

Exponents

• We could rewrite it as:

(4 x 4 x 4) x (4 x 4)

• Or with an exponent:

• Note that we did NOT multiply the exponents, we ADDED them!

43 · 42 =

(32)

Another Example

(33)

Another Example

• We would rewrite the equation as:

• Or more simply as: 32

• Again, we ADDED the exponents

3

-2

·

3

4

1 3 ·

1

3 · 3· 3· 3· 3 =

(34)

Example with a variable

• What is:

(35)

Example with a variable

• What is:

• Rewrite as:

• Or more simply as:

x

3

_·

_x

-4

₌

x· x· x

( ) · 1

x · 1 x · 1 x · 1 x æ è ç ö ø

÷ = x· x· x

x· x· x· x

1

x = x

(36)

Product Law of

Exponents

• General Form:

• If the bases are identical, just add the exponents.

(37)

Contra-Example

• Does the Product Law of Exponents apply here?

(38)

Contra-Example

• Does the Product Law of Exponents apply here?

• No! We do NOT have a common base or a common exponent.

(39)

Product Law of Exponents

• What about:

(40)

Product Law of Exponents

• What about:

• Does it matter what order we write the factors?

(41)

Product Law of Exponents

• What about:

• How about:

2 b

3

·

5 b

-2

=

(42)

Product Law of Exponents

• What about:

• How about:

2 b

3

_·

₅

_b

-2

₌

2 · 5 · b-2 _· _b3 ₌

(43)

More Examples

2 b

·

3 b

=

5 z

2

· ₆

_z

3

=

(44)

More Examples

2 b

·

3 b

=

6 b

2

5 z

2

· ₆

_z

3

=

₃₀

_z

5

(45)

The Other Product Law

• How would we work with:

• What’s different/what’s the same?

(46)

The Other Product Law

• We can rewrite as: (4 x 4) x (3 x 3) • Or as (4 x 3) x (4 x 3) because order

doesn’t matter when we multiply

4

2

·

3

2

=

(47)

The Other Product Law

• We can rewrite as: (4 x 4) x (3 x 3)

• Or as (4 x 3) x (4 x 3) because order doesn’t matter when we multiply

• Now we can rewrite as

4

2

·

3

2

=

4 · 3

(

)

2

(48)

The Other Product Law

• What’s different/what’s the same?

(49)

The Other Product Law

• We can rewrite as:

• (5 x 5 x 5 x 5) x (4 x 4 x 4 x4)

• Or as (5 x 4) x (5 x 4) x (5 x 4) x (5 x 4) because order doesn’t matter when we multiply

5

4

·

4

=

(50)

The Other Product Law

• We can rewrite as:

• (5 x 5 x 5 x 5) x (4 x 4 x 4 x4)

• Or as (5 x 4) x (5 x 4) x (5 x 4) x (5 x 4) because order doesn’t matter when we multiply

• Now we can rewrite as

5

4

·

4

=

5

·

4 (

)

4

(51)

The Other Product Law

• When the exponents are identical, but the bases are different, we MULTIPLY the bases and keep the exponent the same.

(52)

More Examples

• Try these on your own:

4

6

·

6

=

5-11 ·11-11 =

1 4 æ è ç ö ø ÷ 3 · 2 3 æ è ç ö ø ÷ 3 =

(53)

(54)

Quotient Law of Exponents

(55)

Quotient Law

• If we have a common base and are multiplying, we add the exponents. • If we have a common base and are

(56)

Example

(57)

Example

• This could also be written as:

7

6

¸

7

3

=

(58)

Another Example

x

9

(59)

Example

(60)

Example

7

5

¸

7

-3

=

(7 · 7 · 7 · 7 · 7) ¸ (1 7 ·

1 7 ·

(61)

Example

• Using ‘flip and multiply’ we get:

7

5

¸

7

-3

=

(7 · 7 · 7 · 7 · 7) ¸ (1 7 · 1 7 · 1 7)

(62)

Another Example

x

-9

(63)

Quotient Law of Exponents

• General Form:

Or

ab ¸ ac = ab-c

ab

(64)

Now You Try These

x

-4

¸

x

-2

=

(65)

Other Quotient Law

• Same Exponents, but Different Base • The Product Law told us that:

• So what would you do with this:

64 · 34 =184

(66)

Quotient Law

• We can rewrite the problem as:

6

·

6

·

6

·

6

3

·

3

·

3

·

3 =

(67)

Quotient Law

• We can rewrite the problem as:

6

·

6

·

6

·

6

3

·

3

·

3

·

3 =

2

4

(68)

More Examples

32

6

¸

4

6

=

(69)

Quotient Law of Exponents

• General Form:

• Or written another way:

a

c

¸

b

c

=

(

a

¸

b

)

c

a

c

b

c

=

(70)

Coefficients

• What do we do with these:

8 x

6

_·

₄

_x

3

8 x

6

(71)

How About This One?

-

42 x

4

y

3

z

(72)

How About This One?

-

42 x

4

_y

3

_z

7 x

2

_yz

5

= -

6 x

2

_y

2

_z

-4

or

-

42 x

4

y

3

z

7 x

2

_yz

5

=

-6

x

2

y

2

(73)

How about this one?

(74)

Classwork

(75)

Power Law of Exponents

• Key Skill: WWBAT use the Power Law of Exponents to raise a power to a

(76)

Power Law

• How do we read this:

• It can be rewritten as:

5

3

( )

2

=

(77)

Power Law

• Which can be solved TWO ways:

– Use same base:

– Use same exponents:

– Answers are identical as shown by Power Law

5

3

( )

2

=

5

3

·

5

3

=

5

6

(78)

Another Example

• How about:

x4

( )

3

(79)

Another Example

• How about:

x4

( )

3

=

(80)

Negative Exponents

5

-3

( )

2

(81)

Negative Exponents

5

-3

( )

2

=

(82)

Another Example

• How about:

x

4

( )

-3

(83)

Another Example

• How about:

x

4

( )

-3

(84)

Power Law of Exponents

• General Form:

a

b

( )

c

(85)

Interesting Application

• Is there a way to use the Power Law to simplify this?

(86)

Interesting Application

• Is there a way to use the Power Law to simplify this?

• This ONLY works when one base is related to the other by an exponent.

(87)

How about this one?

(88)

How about this one?

5

5 • 3

-

7 =

(89)

Classwork

(90)

4.1.1 - Scientific Notation

(91)

Very Large Numbers

(92)

Scientific Notation

(93)

The Powers of 10

• What happens when we raise the

(94)

Powers of 10

102 =

103 =

106 =

(95)

Powers of 10

102 = 100

103 = 1,000

106 = 1,000,000

(96)

Powers of Ten

• http://www.youtube.com/watch?v=0f

(97)

Combine through

Multiplication

• What if we took a number and multiplied it by a power of 10?

5 x 103

(98)

Combine through

Multiplication

• What if we took a number and multiplied it by a power of 10?

5 x 103 = 5 x 1,000 or 5,000

(99)

Find the value of ‘n’

10n = 10,000

103 = n

(100)

Try on Your Own

• Find each Product:

8 x 105

1,620 x 105

14.9 x 103

0.57 x 106

• Find the value of ‘n’:

6 x 10n = 600

54 x 10n = 5,400,000

n x 102 = 800

(101)

Key Vocabulary

(102)

Key Vocabulary

(103)

Examples

• 2.99 x 109 and 9.4 x 1011 are both in

Scientific Notation

• 11 x 108 and 0.44 x 102 are NOT in

(104)

Key Vocabulary

• Standard Notation is a number written without any exponents.

(105)

Classwork

(106)

4.1.2 - Converting to

Scientific Notation

(107)

Key Vocabulary

(108)

Key Vocabulary

(109)

Conversions

• Take the number 18 x 105

(110)

More Examples

• Convert the following into SNOT

13 x 108

102 x 103

(111)

More Examples

• Convert the following into SNOT

13 x 108 = 1.3 x 109

102 x 103 = 1.02 x 105

(112)

Classwork

• Purple Books!

(113)

4.2.3 - Scientific Notation

w/Negative Exponents

(114)

SNOT with Negatives

• Let’s look at this: 3 x 10-4

(115)

SNOT with Negatives

• Let’s look at this: 3 x 10-4

• How would we rewrite this using the “elevator”?

• 3 divided by 104 = 3 divided by 10,000

or 0.0003

3 ´ 1

104 =

3

(116)

SNOT with Negative

• To change a very small number from standard notation to Scientific

Notation, we:

1) Move the decimal so that we have a number ≥ 1 and < 10.

2) We multiply that number by 10 raised to a

(117)

Examples

0.00375 =

0.00005 =

(118)

Examples

0.00375 = 3.75 ´10-3

0.00005 = 5 · 10-5

(119)

Classwork

• Purple Books

(120)

4.1.4 - Exponent Laws

and Scientific Notation

• Key Skill: WWBAT use the Exponent Laws in working with Scientific

(121)

Key Vocabulary

• Scientific Notation is a form in which a number is expressed as the product of a number greater than or equal to 1 and less than ten, and a power of 10

(122)

Examples

• Scientific Notation:

• Standard Notation: 3,400,000

• Neither:

3.4 ´

10

6

(123)

Scientific Notation

• Scientific Notation can also be used to describe very small numbers.

(124)

For Example

0.0000563

=

5.63 ´

10

-5

0.0000563 = 563

10,000,000 563 10,000,000 = 563 107 563

107 = 563 ´10

-7

(125)

Or more simply…..

• Count the decimal places and use a negative exponent

(126)

Examples

• Covert the following to scientific notation:

0.0023

(127)

Examples

• Covert the following to scientific notation:

0.0023 = 2.3 x 10-3

(128)

Examples

• Covert the following to standard notation:

2.4 ´

10

-3

(129)

Examples

• Covert the following to standard notation:

2.4 ´

10

-3

=

0.0024

(130)

(131)

Stars in the Universe

• If there are approximately 2 x 1011

stars in a galaxy and there are

approximately 2 x 1011 galaxies, how

(132)

Solution

2 x 1011 x 2 x 1011 = set up problem

2 x 2 x 1011 x 1011 reorder

(133)

Using the Product Law

with Scientific Notation

• If one star is 7 x 1016 miles away from

earth, and another star is 3 x 106 times

(134)

Using the Product Law

with Scientific Notation

7 x 1016 x 3 x 106 = set up problem

7 x 3 x 1016 x 106 reorder

21 x 1022 multiply through

2.1 x 1023 change to

(135)

Using the Quotient Law

with Scientific Notation

• If there are 8.8 x 1014 grains of rice in a

harvest and the population of the city is 2.2 x 104, how many grains of rice

(136)

Using the Quotient Law

with Scientific Notation

8.8 x 1014 ÷ 2.2 x 104 set up problem

use division bar

4 x 1010 divide numbers

subtract exponents

(137)

Quotient Law

• If a bacteria cell measures 2.4 x 10-8

millimeters and a virus measures 8 x 10-17 millimeters, how many times

(138)

Quotient Law

• Solution is shown below. Make sure do

deal with negative numbers correctly and to give your answer in Scientific Notation.

2.4 ´

10

-8

8 ´

10

-17

=

0.3 ´

10

(139)

Classwork

(140)

4.14 - Operations with

Scientific Notation

(141)

Addition

(142)

Addition

• How would we add two numbers like: 2 x 103 and 4 x 103

• Let’s rewrite them in standard

notation: 2,000 + 4,000 = 6,000 or in scientific notation: 6 x 103

(143)

Subtraction

(144)

Subtraction

• What is 5 x 104 - 3 x 104 ?

• Again, we could rewrite the problem as 50,000 - 30,000 = 20,000 and then

restate in scientific notation as 2 x 104

(145)

Multiplication

• If you double 4 x 104, what would you

(146)

Multiplication

• If you double 4 x 104, what would you

get?

• We could rewrite as 2 x 40,000 = 80,000 and change it back to 8 x 104

• Or we could rewrite this as: 2 x 4 x 104 = 8 x 104

(147)

Division

(148)

Division

• What is half of 6 x 103 ?

• Again, we could change to standard

notation and 6,000 divided by 2 = 3,000 or 3 x 103

• Note again, the exponent is

(149)

When Exponent IS Affected

(150)

When Exponent IS Affected

• Sometimes the exponent DOES change. • Try: 4 x 103 + 6 x 103

• We have 4,000 + 6,000 = 10,000 or 10 x 103

• To get back to SNOT, we would have to change our answer to 1 x 104

• The exponent only changes because we went over 10, violating the SNOT

(151)

Challenge Question

• How would we solve this:

(152)

Classwork

5.2 ´

10

7

+

3.6 ´

10

7

=

7.8 ´

10

12

+

4.5 ´

10

12

=

8.0 ´

10

16

-

3.5 ´

10

16

=

4.2 ´

10

9

-

3.5 ´

10

9

=

(153)

Classwork

5.2 ´

10

7

+

3.6 ´

10

7

=

8.8 ´

10

7

7.8 ´

10

12

+

4.5 ´

10

12

=

1.23 ´

10

13

8.0 ´

10

16

-

3.5 ´

10

16

=

4.5 ´

10

16

4.2 ´

10

9

-

3.5 ´

10

9

=

7 ´

10

8

(154)

4.2.1 - Exponential

Relationships

(155)

Exponent Example

(156)

Comparing Linear and

Nonlinear Relationships

• How are these two equations different? • How will they look differently on a

graph?

y

=

x

(157)

Classwork

(158)

4.2.2 - Exponential

Growth

• Key Skill: WWBAT recognize

(159)

Constant versus

Exponential Growth

• Compare these two tables:

x 1 2 3 4 5

y 2 4 6 8 10

x 1 2 3 4 5

(160)

(161)

Key Vocabulary

• Exponential growth is found when

quantities are repeatedly multiplied by a number greater than 1.

(162)

Exponential Growth Factor

(163)

Key Vocabulary

• Exponential Growth Factor is a

number greater than 1 that is used to multiply a prior value to arrive at the next value.

(164)

Multiple Years

(165)

Multiple Years

• To find the size of our bank account in 10 yrs, we can multiple $1,000 by 1.03 10 times. But how could we do it more quickly?

• 1.0310 x 1000 = $1,343.92

(166)

“Rule of 72”

• If I earn 7% per year on $1,000, in how many years will I have $2,000?

(167)

Classwork

(168)

4.2.3 - Exponential Decay

(169)

Constant versus

Exponential Decay

• Compare these two tables:

x 0 1 2 3 4

y 8 7 6 5 4

x 0 1 2 3 4

(170)

(171)

Exponential Decay

• A rubber ball will bounce to a height equal to 60% of the height from which it is dropped. If the ball is dropped

(172)

Key Vocabulary

• Exponential Decay is seen when

quantities are repeatedly multiplied by a number between 0 and 1.

(173)

Key Vocabulary

• Exponential Decay Factor is a number between 0 and 1 multiplied by each

(174)

Bouncing Ball Problem

• Our equation is:

• What is the “decay factor” in that equation?

(175)

Superball Problem

• Our equation is:

• What is the “decay factor” in that equation?

• The decay factor is 0.8

(176)

Classwork

(177)

4.2.3 - More Exponential

Growth/Decay

• Key Skill: WWBAT calculate

(178)

Exponential Growth or Decay?

y = .8x

y = 1.04x

y = .5x

y = 3x

y

=

3

4 æ

è

ç

ö

ø

÷

x

y

=

4

(179)

Calculating Decay Factors

(180)

Calculating Decay Factors

• A $20,000 car depreciates by 25% per year. What is its decay factor?

• The decay factor is NOT 25%. We want the car to be worth $15,000 after 1

(181)

Another Example

(182)

Another Example

• A special medicine in the bloodstream declines by 4% each hour. What is its decay factor?

(183)

Exponential Growth

• 100 snakes are released into the wild to control a local rodent population.

• With much food, the snake population grows at 20%/year.

#1) What is the growth factor?

(184)

Exponential Growth

• 100 snakes are released into the wild to control a local rodent population.

• With much food, the snake population grows at 20%/year for 3 years.

#1) What is the growth factor? 1.2

#2) Snake count after 3 years? 5 years? Write an expression and two totals.

(185)

Exponential Decay

• A radioactive element has a half-life of 1 year. There are 100 pounds of the

element in a box.

#1) What is the decay factor?

#2) Write expressions to show how many

(186)

Exponential Decay

• A radioactive element has a half-life of 1 year. There are 100 pounds of the element in a box. #1) What is the decay factor? 0.5

#2) Write expressions to show how many pounds are left after 3 yrs and 5 yrs, then calculate how many pounds are left.

100 x 0.53 or 100 x 0.5 x 0.5 x 0.5 = 12.5

100 x 0.55 or 100 x 0.5 x 0.5 x 0.5 x 0.5 x 0.5=

(187)

Summary

• We often must ADD a percentage

increase to the number 1 to get the Growth Factor.

• We must often SUBTRACT a

(188)

Classwork

(189)

4.3.1 - Radicals

• Key Skill: WWBAT identify both a

(190)

Key Vocabulary

• A square root asks for a number or

numbers that when multiplied together amount to a given value.

• We call this sign a “radical sign”.

(191)

(192)

Perfect Squares

• Perfect squares are numbers that turn into whole numbers when you take

their square root.

(193)

Perfect Square Curiosity

• 1+3 = 4 • 1+3+5 = 9

• 1+3+5+7 = 16 • 1+3+5+7+9 = 25

• 1+3+5+7+9+11 = 36

(194)

Not Perfect Squares, but…

• Only integers are perfect squares, but it can be simple to find the square

roots of numbers like these:

(195)

Not Perfect Squares, but…

• Only integers are perfect squares, but it can be simple to find the square

roots of numbers like these:

– 1/4, 1/9, 9/16, 4/25 = 1/2, 1/3, 3/4, 2/5

(196)

Imperfect Squares

• When we take the square root of an

imperfect square, we can estimate the answer by finding perfect squares

NEAR the imperfect square.

(197)

Other Roots

(198)

Negative Square Roots

• We usually indicate a negative root by the following:

(199)

(200)