8th Grade Slides Exponents.

8th Grade Slides Exponents

Exponents Key Skill: Calculate values of expressions with positive and negative exponents.

Quick Review Exponent Base

Exponent Review Repeated multiplication:

Exponent Review Any number to the first power is?

Exponent Review Any number to the first power is itself So And

Exponent Review Any non-zero number to the power of 0 is?

Exponent Review Any non-zero number to the power of 0 is 1. So,
and (assuming x does not = 0),

Question What is

Question What is Is it the same as

Odd and Even Exponents Will a negative number raised to a power always be positive?

Odd and Even Exponents How do we know whether a negative number raised to a particular power will be positive or negative?

Odd and Even Exponents How do we know whether a negative number raised to a particular power will be positive or negative? Negative base with an odd exponent results in a NEGATIVE answer Negative base with an even exponent results in a POSITIVE answer

Decimals and Fractions
Is this the same as

Decimals and Fractions
Is this the same as No!

Negative Exponents Look at this progression of numbers: 33 = 32 = 31 =
30 = 3-1 = 3-2 = 3-3 =

Negative Exponents Look at this progression of numbers: 33 = 27 32 = 9
31 = 3 30 = 1 3-1 = 1/3 3-2 = 1/9 3-3 = 1/27

Negative Exponents Rule:

Negative Exponents The reverse is also true:
Write in numerical example

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 denominator, we move it to the numerator.

Method Step 1: Take the RECIPROCAL of the base.
What does RECIPROCAL MEAN? Step 2: The exponent moves with the base, but the negative sign in the exponent disappears.

Examples

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

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)

Classwork Pages

Exponent Laws Key Skill: Apply the Laws of Exponents to simplify expressions.

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

PEMDAS Parentheses ( ) Exponents 3 Multiplication x, Division ÷
Addition + Subtraction __

P lease Excuse My Dear Aunt Sally Parentheses ( ) Exponents 3
Multiplication x, Division ÷ Addition + Subtraction __ P lease Excuse My Dear Aunt Sally

Product Law of Exponents
How would we multiply:

How would we multiply: We could rewrite it as: (4 x 4 x 4) x (4 x 4)

How would we multiply: 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!

Another Example How would we multiply:

Another Example How would we multiply:
We would rewrite the equation as: Or more simply as: 32 Again, we ADDED the exponents

Example with a variable
What is: Write answer on board after pause.

Example with a variable
What is: Rewrite as: Or more simply as:

Contra-Example Does the Product Law of Exponents apply here?

Contra-Example Does the Product Law of Exponents apply here?
No! We do NOT have a common base or a common exponent.

What about:

What about: Does it matter what order we write the factors?

What about: Does it matter what order we write the factors? How about:

More Examples

The Other Product Law How would we work with:
What’s different/what’s the same?

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

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

What’s different/what’s the same?

The Other Product Law How would we work with: 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

The Other Product Law How would we work with: 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

The Other Product Law When the exponents are identical, but the bases are different, we MULTIPLY the bases and keep the exponent the same.

More Examples Try these on your own:

Classwork

Quotient Law of Exponents
Key Skill: WWBAT use the quotient law of exponents to simplify expressions.

Quotient Law If we have a common base and are multiplying, we add the exponents. If we have a common base and are dividing, what might we do?

Example

Example This could also be written as:

Another Example

Example

Example This could also be written as:

Example This could also be written as:
Using ‘flip and multiply’ we get:

Another Example

General Form: Or

Now You Try These

Other Quotient Law Same Exponents, but Different Base
The Product Law told us that: So what would you do with this:

Quotient Law So what would you do with this:
We can rewrite the problem as:

More Examples

General Form: Or written another way:

Coefficients What do we do with these:

How About This One?

How About This One? or

How about this one?

Classwork Page 154 #4

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

Power Law How do we read this: It can be rewritten as:

Power Law How do we read this: It can be rewritten as:
Which can be solved TWO ways: Use same base: Use same exponents: Answers are identical as shown by Power Law

Another Example How about:

Another Example How about: It can be rewritten as:

Negative Exponents How do we read this:

Negative Exponents How do we read this: It can be rewritten as:

Another Example How about:

Power Law of Exponents General Form:

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

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.

How about this one?

How about this one? 5 and 3 are not related by any exponent,
so there is nothing to do

Classwork Page 155 #6-8

Scientific Notation Key Skill: WWBAT multiply number by powers of 10 AND recognize if numbers are in scientific notation.

Very Large Numbers How many stars are there in the sky?

Scientific Notation Is there an easier way to write very large numbers?

The Powers of 10 What happens when we raise the number 10 to ever larger exponents?

Powers of 10 102 = 103 = 106 = 109 =

Powers of 10 102 = 100 103 = 1,000 106 = 1,000,000 109 = 1,000,000,000

Powers of Ten

Combine through Multiplication
What if we took a number and multiplied it by a power of 10? 5 x 103 8.7 x 105

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 8.7 x 105 = 8.7 x 100,000 or 870,000

Find the value of ‘n’ 10n = 10,000 103 = n n x 106 = 4,600,000

Try on Your Own Find each Product: 8 x 105 1,620 x 105 14.9 x 103
Find the value of ‘n’: 6 x 10n = 600 54 x 10n = 5,400,000 n x 102 = 800 9.8 x n = 9,800,000

Key Vocabulary Scientific Notation

Key Vocabulary Scientific Notation is a value expressed as a product of a number ≥ 1 and < 10 and a power of 10

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 Scientific Notation

Key Vocabulary Standard Notation is a number written without any exponents. Numbers like 3 or 14,000 or are said to be in ‘standard notation’.

Classwork Purple Books! Page 180 #1-15

4.1.2 - Converting to Scientific Notation
Key Skill: WWBAT change numbers into proper SNOT form.

Key Vocabulary Googol

Key Vocabulary Googol is 10100

Conversions Take the number 18 x 105
How would we write this number in proper Scientific Notation?

More Examples Convert the following into SNOT 13 x 108 102 x 103

More Examples Convert the following into SNOT 13 x 108 = 1.3 x 109

Classwork Purple Books! Page #15-21

4.2.3 - Scientific Notation w/Negative Exponents
Key Skill: WWBAT convert between standard numbers and SNOT with negative exponents.

SNOT with Negatives Let’s look at this: 3 x 10-4
How would we rewrite this using the “elevator”?

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

SNOT with Negative To change a very small number from standard notation to Scientific Notation, we: Move the decimal so that we have a number ≥ 1 and < 10. We multiply that number by 10 raised to a negative exponent equal to the number of decimal places we have moved.

Examples

Classwork Purple Books Pages #1-12 and 28-31

4.1.4 - Exponent Laws and Scientific Notation
Key Skill: WWBAT use the Exponent Laws in working with Scientific Notation.

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 Standard Notation is a number written without exponents

Examples Scientific Notation: Standard Notation: 3,400,000 Neither:

Scientific Notation Scientific Notation can also be used to describe very small numbers. Negative exponents are used in scientific notation.

For Example

Or more simply….. Count the decimal places and use a negative exponent

Examples Covert the following to scientific notation: 0.0023

Examples Covert the following to scientific notation:

Examples Covert the following to standard notation:

2 x 1011 stars in a galaxy

Stars in the Universe If there are approximately 2 x 1011 stars in a galaxy and there are approximately 2 x 1011 galaxies, how many stars exist?

Solution 2 x 1011 x 2 x 1011 = set up problem
2 x 2 x 1011 x 1011 reorder 4 x multiply through

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 FURTHER away, how far is the second star from earth?

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 multiply through 2.1 x change to Scientific Notation

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 (split evenly) would each person get?

Using the Quotient Law with Scientific Notation
8.8 x 1014 ÷ 2.2 x 104 set up problem use division bar 4 x divide numbers subtract exponents

Quotient Law If a bacteria cell measures 2.4 x 10-8 millimeters and a virus measures 8 x millimeters, how many times larger is the bacteria cell?

Quotient Law Solution is shown below. Make sure do deal with negative numbers correctly and to give your answer in Scientific Notation.

Classwork Page #1-5

4.14 - Operations with Scientific Notation
Key Skill - WWBAT add and subtract numbers in Scientific Notation.

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

Addition How would we add two numbers like: 2 x 103 and 4 x 103
Let’s rewrite them in standard notation: 2, ,000 = 6,000 or in scientific notation: 6 x 103 Note that we did NOT need to add exponents to get our answer!

Subtraction What is 5 x x 104 ?

Subtraction What is 5 x 104 - 3 x 104 ?
Again, we could rewrite the problem as 50, ,000 = 20,000 and then restate in scientific notation as 2 x 104 Again, note that we did NOT need to change the exponent.

Multiplication If you double 4 x 104, what would you get?

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: x 4 x 104 = 8 x 104 Note again, the exponent in unchanged.

Division What is half of 6 x 103 ?

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 unaffected.

When Exponent IS Affected
Sometimes the exponent DOES change. Try: 4 x x 103

When Exponent IS Affected
Sometimes the exponent DOES change. Try: 4 x x 103 We have 4, ,000 = 10,000 or 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 format

Challenge Question How would we solve this:

Classwork

4.2.1 - Exponential Relationships
Key Skill: WWBAT define and compare exponential relationships.

Exponent Example Folding paper

Comparing Linear and Nonlinear Relationships
How are these two equations different? How will they look differently on a graph?

Classwork Pages 170 #1-7

Exponential Growth Key Skill: WWBAT recognize relationships that exhibit exponential growth and their growth factors.

Constant versus Exponential Growth
Compare these two tables: x 1 2 3 4 5 y 6 8 10 x 1 2 3 4 5 y 8 16 32

y=2x versus y=2x

Key Vocabulary Exponential growth is found when quantities are repeatedly multiplied by a number greater than 1. “Increasing at an increasing rate”

Exponential Growth Factor
If I have $1,000 in the bank, growing at 3% per year, what number do I multiply the 1,000 by to get next year’s TOTAL account value?

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. In the prior example, the growth factor is 1.03

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

Multiple Years To find the size of our bank account in 10 yrs, we can multiple $1,000 by times. But how could we do it more quickly? x 1000 = $1,343.92 The works because exponents indicate repeated multiplication.

“Rule of 72” If I earn 7% per year on $1,000, in how many years will I have $2,000? If I earn 10% per year, in how many years will I double my money?

Exponential Decay Key Skill: WWBAT identify exponential decay and decay factors.

Constant versus Exponential Decay
Compare these two tables: x 1 2 3 4 y 8 7 6 5 x 1 2 3 4 y 8 0.5

y=-x+8 versus y=8·0.5x

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 from a height of 10 feet how high will it be on the 5th bounce?

Key Vocabulary Exponential Decay is seen when quantities are repeatedly multiplied by a number between 0 and 1. “decreasing at a decreasing rate”

Key Vocabulary Exponential Decay Factor is a number between 0 and 1 multiplied by each prior value to calculate the next value.

Bouncing Ball Problem Our equation is:
What is the “decay factor” in that equation?

Superball Problem Our equation is:
What is the “decay factor” in that equation? The decay factor is 0.8

Classwork Pages #1-5

4.2.3 - More Exponential Growth/Decay
Key Skill: WWBAT calculate exponential growth and decay factors.

Exponential Growth or Decay?
y = .8x y = 1.04x y = .5x y = 3x

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

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 year, so we must use a multiplier of = 0.75

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

Another Example A special medicine in the bloodstream declines by 4% each hour. What is its decay factor? The decay factor is = .96

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

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. 100 x 1.23 or 100 x 1.2 x 1.2 x 1.2 = 173

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 pounds are left after 3 yrs and 5 yrs, then calculate how many pounds are left.

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= 3.125

Summary We often must ADD a percentage increase to the number 1 to get the Growth Factor. We must often SUBTRACT a depreciation rate from 1 to get the Decay Factor.

Classwork Page #6-10

Radicals Key Skill: WWBAT identify both a positive and negative square root for a number.

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”.

Examples

Perfect Squares Perfect squares are numbers that turn into whole numbers when you take their square root. Examples: 4, 9, 16, 25, 36, etc.

Perfect Square Curiosity
1+3 = 4 1+3+5 = 9 = 16 = 25 = 36 = 49

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 0.04, 0.25, 0.36, 0.49

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 0.04, 0.25, 0.36, 0.49 = 0.2, 0.5, 0.6, 0.7

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. What integer is the square root of 37 near?

Other Roots Is there a number OTHER THAN 5, that when squared is equal to 25?

Negative Square Roots We usually indicate a negative root by the following:

Examples

Operations with Radicals

Rules for Radicals Radicals are the same as exponents, in fact the square root of 2 is equal to 21/2 True with variables as well:

Rules for Radicals Radicals are the same as exponents, in fact the square root of 2 is equal to 21/2 When you add and subtract radicals, the radicals DO NOT CHANGE, just as with exponents When you multiply and divide radicals, they DO CHANGE, just as with exponents

More Examples

Classwork p #1-22

Solving Equations with Radicals
Key Skill: WWBAT solve equations with radicals.

Graph of Square Roots What does the graph of this equation look like?

Solving Equations What is the opposite of taking the square root of a number?

Solving Equations What is the opposite of taking the square root of a number? Squaring the number! So:

Solving Equations

4.3.2 - Simplifying Radicals
Key Skill: WWBAT simplify a radical expression.

Square Roots Are these square roots equal?

Factoring Square Roots
We can use this property of square roots to simplify problems Example: How could we simplify the following:

Process Step One: Examine the factors of the number under the radical sign. Step Two: Pick the pair of factors with the LARGEST perfect square. Step Three: Take the square root of the perfect square and write it OUTSIDE the radical sign, leaving the other factor INSIDE the radical sign.

More Examples Simplify the following:

Factoring with Variables
How could you factor this to simplify it?

More Examples Simplify the following:

Simplified Radicals Key Skill: WWBAT simplify radicals and use them in operations.

Can we add these? Just as we cannot add x2 + x, we also cannot add

How about this one?

Try These Use KUTA worksheet problems

nth Roots Key Skill: WWBAT solve for roots other than square roots.

Exponents and Opposites
Look at these exponents: x n7 If the way to solve for ‘x’ in x2 = 100 is to take the square root of both sides, what would we do to solve x3 = 64 ?

Exponents and Opposites
Look at these exponents: x n7 If the way to solve for ‘x’ in x2 = 100 is to take the square root of both sides, what would we do to solve x3 = 64 ? The “Cube Root” of 64

Cube Roots When we take the square root, we ask what times itself equals our number. With a cube root, we ask what times itself times itself again equals our number.

Cube Root Examples

Odd and Even Roots You CAN take Odd roots of negative numbers
You CANNOT take Even roots of negative numbers OK Not OK

Other Roots We do not have to stop at cube roots:

Classwork Pages , #1-20

8th Grade Slides Exponents.

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