im 8 ch 8.2.3 what happens if the exponent is...

IM 8 Ch 8.2.3 What Happens If The Exponent Is Negative

CPM Materials modified by Mr. Deyo

What part of the expression is being raised to a power?

What are the factors?

How can we rewrite the expression to have fewer

terms?

Common Core Standard: 8.EE.1


By the end of the period, I will develop methods for simplifying expressions with positive exponents, and will explain what negative and zero exponents represent.

I will demonstrate this by completing Four‑Square notes and by solving problems in a pair/group activity.

Learning TargetTitle: IM8 ‑ Ch. 8.2.3 What Happens If The Exponent Is Negative? Date:


Home Work: Sec. 8.2.3Desc. Date Due

Review & Preview

Day 1: 4 Problems 8‑88, 8‑90, 8‑93, 8‑94

Day 2: 3 Problems 8‑95, 8‑96, 8‑98


Vocabulary1) Base

2) Exponent

3) Zero Exponent

4) Negative Exponent


8.2.3 What Happens If The Exponent Is Negative?Earlier in this chapter you learned how to write large numbers in scientific notation. Astronomers use those large numbers to measure great distances in space. Not all scientists work with such a large scale, however. Some scientists use very small numbers to describe what they measure under a microscope. In this lesson, you will continue your work with exponents, and then you will turn your attention to using scientific notation to represent small numbers.

What part of the expression is being raised to a power?

What are the factors?

How can we rewrite the expression to have fewer

terms?


872. Two of the problems below are correct, and four contain errors. Expand each original expression to verify that it is correct. If it is not, identify the mistake and simplify to find the correct answer.

a)

b)

c)

d)

e)

f) z3

z3= 0

62

65= 63

?

?

83

82= 8

7?72

37

35 32 = 97?x3y2x4 = x7y2?

23 24 = 212?

.

.

.

.


873. Rewrite each expression in a simpler form using the patterns you have found for rewriting expressions with exponents. If it is reasonable, write out the factored form to help you.

a) b)

c) d)

23 24

5x2( )3

28

25.

4x 2( ) 5x2( )


You have worked with your team to describe ways to rewrite and simplify exponent expressions involving multiplication and division. In your Learning Log, give examples of each kind of expression. Describe in words how to rewrite each one. Include factored and simplified expressions to go with your descriptions.

874. LEARNING LOG “Simplifying Exponent Expressions” Date_______


875. Salvador was studying microscopic pond animals in science class. He read that amoebas were 0.3 millimeters to 0.6 millimeters in length. He saw that euglenas are as small as 8.0 · 10−2 millimeters, but he did not know how big or small a measurement that was. He decided to try to figure out what a negative exponent could mean.

a) Complete Salvador’s calculations. Use the pattern of dividing by 10 to fill in the missing values.

b) How does 102 related to 10–2?

103 = 1000 10102 = 100 10101 = 10 10100 = 101 =102 =103 =

c) What type of numbers did the negative exponents create?

Did negative exponents create negative numbers?


876. Ngoc was curious about what Salvador was doing and began exploring patterns, too. He completed the following calculations.

a) Complete his calculations. Be sure to include all of the integer exponents from 5 to –3. 25 = 32

24 = 1623 = 822 = 21 = 20 = 21 =

22 =

23 =

b) Look for patterns in his list. How are the values to the right of the equal sign changing?

Is there a constant multiplier between each value?

12


877. In problems 875 and 876, you saw that 100 and 20 both simplify to the same value. What is it? Do you think that any number to the zero power would have the same answer? Explain.


878. Both Ngoc and Salvador are looking for ways to calculate values with negative exponents without extending a pattern. Looking at the expression 5−2, they each started to simplify differently.

Salvador thinks that 52 is 25, so 5−2 must be .

Ngoc thinks 5−2 is .

Which student is correct?

Why?

1 25

1 52


879. Salvador’s first questions about negative exponents came from science class, where he had learned that euglenas measured 8.0 × 10−2 millimeters. Use your understanding of negative exponents to rewrite 8.0 × 10−2 in standard form.


880. The probability of being struck by lightening in the United States is 0.000032%, while the probability of winning the grand prize in a certain lottery is 6.278 × 10−7 percent. Which event is more likely to happen? Explain your reasoning, including how you rewrote the numbers to compare them.


881. The list below contains a star, a river, and a type of bacteria. Which one is which? Use your understanding of scientific notation to put the three items listed below in order from largest to smallest and identify which is which.

Yangtze measures 6.3 × 105 meters.

Staphylococcus measures 6.0 × 10−7 meters.

Eta Carinae measures 7.1 × 1017 meters.

Explain your reasoning:


a)

c)

882. Which of the numbers below are correctly written in scientific notation? For each that is not, rewrite it correctly.

4.51 × 10−2 b)

d)

0.789 × 105

31.5 × 102 3.008 × 10−8


a)

c)

883. Create a fraction from these expressions, and then show how you use a Giant One to simplify.

65 · 6−3 b)

*d)

w5 w−2

105 w−3 w4·105104 · 105


a)

c)

884. Now it is time to reverse your thinking. If negative exponents can create fractions, then can fractions be written as expressions with negative exponents? Simplify the expressions below. Write your answer in two different forms: as a fraction and as an expression with a negative exponent.

63

65

b)

*d)

m5

m6

63m5

65m6 102

105


a)

c)

885. Rewrite each expression in a simpler form. Visualize the factored form and the Giant One to help you, or write it out if it is reasonable to do so.

(5x2)3

b)

(4x)2(5x2)

a2c11

a5c3


886. Lamar has a list of 600 pages to read to finish his book for his upcoming book report. He sets a goal to read half of the pages left in his book each day.a) How many pages will Lamar have to read the first day to meet his goal?

How many will he have left to read after 3 days?

b) How many pages will he need to read on day 8?

With your team, write two different expressions to show how you found your answer.


887a,b. Additional Challenge: Alice was simplifying the expressions at right when she noticed a pattern. Each exponent under the radical (square root) sign was two times the exponent of the final answer. She wanted to know more about why this was happening.

a) Confirm Alice’s pattern by simplifying the expression on your calculator. Can you rewrite your answer as 5 raised to a power?

b) What does the operation “square root” do to an expression?

What operation does it “undo”?


887c. Additional Challenge: Alice was simplifying the expressions at right when she noticed a pattern. Each exponent under the radical (square root) sign was two times the exponent of the final answer. She wanted to know more about why this was happening.

c) Investigate this pattern using the expression: Expand the expression.

Rewrite 46 as an expression raised to the second power.

Use the square root to “undo” the squaring.

Write your final answer, and check it on your calculator.


887d. Additional Challenge: Alice was simplifying the expressions at right when she noticed a pattern. Each exponent under the radical (square root) sign was two times the exponent of the final answer. She wanted to know more about why this was happening.

d) Use that thinking to rewrite each expression here.


a)

c)

888. Decide which numbers below are correctly written in scientific notation. If they are not, rewrite them.

92.5 × 10−2 b)

d)

6.875 × 102

2.8 × 10 0.83 × 1002

http://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch8/lesson/8.2.3/problem/888

http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CC3/chapter/Ch8/lesson/8.2.3/problem/8-88


889. In the table below, write each power of 10 as a decimal and as a fraction.

a) Describe how the decimals and fractions change as you progress down the table.

b) How would you tell someone how to write 10−12 as a fraction? (You do not have to write the actual fraction.)




890. Mary wants to have $8500 to travel to South America when she is 21. She currently has $6439 in a savings account earning 4% annual compound interest. Mary is 13 now.a) If Mary does not take out or deposit any money, how much money will Mary have when she is 15?

b) Will Mary have enough money for her trip when she is 21?


c) If Mary were to graph this situation, describe what the graph would look like.



a)

c)

891. Recall that vertical lines around a number are the symbol for the absolute value of a number. Simplify each expression.

b)

d)

f)

e)


6

17

4.5

2 5

2 3 5.

2 2



892a,b. For each equation below, solve for x. Sometimes the easiest strategy is to use mental math.

a) x = 1 b) 5.2 + x = 10.95


35

25



892c,d. For each equation below, solve for x. Sometimes the easiest strategy is to use mental math.

d) c) 2x − 3.25 = 7.15


x 16

38=



893. Determine the coordinates of each point of intersection without graphing.

a) y = 2x − 3y = 4x + 1


b) y = 2x − 5y = −4x − 2



a)

c)

894. Write each number in scientific notation.

b)

5467.8


0.0032

8,007,020



a)

c)

895. Simplify each expression using the rules for exponents.

35

310b)

d)

10x4(10x)−2

( )3 · (4)2


14

(xy)3

xy3



896a,b. Simplify each expression. b) a) 1

5 2 15+ ( ) ( )


95

8 15

. 49



896c,d. Simplify each expression. d) c)


48

37

. 35

27

( ) + ( )( ) 3 10

. 57

. 25

( )



896e,f. Simplify each expression. f) e)

12

15

.


2 419

56+8 3



897. Athletes in the Middle Plains School District regularly receive personal advising on their nutrition. Coaches wondered if the nutritional advising was having an impact, so they divided athletes into two groups. One group received advice and one group did not. After six months, they collected the following data:

a) Which is the independent variable?

b) Make a relative frequency table.


c) Is there an association between receiving the nutritional advice and regularly eating a balanced breakfast?



898. Graph the points X(−3, 5), Y(−2, 3), and Z(−1,4). Connect them to make a triangle.a) Reflect the triangle across the y‑axis. What are the new coordinates of point Z?

X'( , ) Y'( , )Z'( , )

b) Translate the original triangle down 6 units and right 3 units. What are the new coordinates of point Y?

c) Dilate the original triangle by multiplying each coordinate by −1. What are the new coordinates?

X"'( , ) Y"'( , )Z"'( , )

X''( , ) Y''( , )Z''( , )

https://www.desmos.com/calculator/lywj1jel5ehttp://homework.cpm.org/cpmhomework/homework/category/CC/textbook/CC3/chapter/Ch8/lesson/8.2.3/problem/898

https://www.desmos.com/calculator/lywj1jel5e



899. The table below shows the amount of money Francis had in his bank account each day since he started his new job.

a) Write a rule for the amount of money in Francis’s account. Let x represent the number of days and y represent the number of dollars in the account.

b) When will Francis have more than $1000 in his account?

https://www.desmos.com/calculator/yh7cv7jt4x


y = ( )x + ( )

Francis will have more than $1000 in his account after ______ days.

https://www.desmos.com/calculator/yh7cv7jt4x


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