Exponentsjtownsendmath.weebly.com/uploads/8/1/8/3/81834070/...You can use what you learned about multiplying numbers with powers to find a shortcut for simplifying expressions with

Unit 14 CCM6+/7+ Page 1

Page 1

UNIT 14

Exponents

and

Scientific Notation

CCM6+/7+

Name_________________________________

Math Teacher___________________________

Projected Test Date___________

Main Ideas Page(s) Unit 14 Vocabulary 2

Exponent Basics, Zero & Negative Exponents 3 – 6

Multiplying, Dividing, and Raising a Power to a Power 7 – 13

Laws of Exponents Review 14 – 15

Intro to Scientific Notation 16 – 19

Operations with Scientific Notation 20 – 22

Real World Scientific Notation Problems 23 – 24

Study Guide 25 – 28

http://cmapp.wcpss.net/uploads/files/2013_middle_school_math/ccm7_plus/ccm7_plus_unit10/day_130_review_of_perimeter,_circumference,_and_area_ccm7plus_unit10.docx


Page 2

CCM6+/7+ Plus Unit 14: Exponents and Scientific Notation

Base When a number is raised to a power, the number that is used as a factor.

Dividing Powers with

the Same Base Property For every nonzero number a and integers m and n,

Exponent The number that indicates how many times the base is used as a factor.

Exponential Form A number is written in exponential form when it has a base and an exponent.

Irrational Numbers A number that cannot be expressed as a ratio of two integers (or as a repeating or

terminating decimal)

Laws of Exponents The "Laws of Exponents" (also called "Rules of Exponents") come from three

ideas: The exponent says how many times to use the number in a multiplication.

Multiplication Property

of Exponents

For any nonzero number a and integers m and n, am•an=am+n

Perfect Cube The cube of a rational number

Perfect Square The square of a rational number

Power The power of a number says how many times to use the number in a

multiplication.

Raising a Power to a

Power Property

For every nonzero number a and integers m and n, (am)n=amn

Raising a Product to a

Power Property

For every nonzero number a and b integer n, (ab)n=anbn

Raising a Quotient to a

Power Property For every nonzero numbers a and b and integer n,

Rational Numbers A number expressible in the form of a/b or -a/b for some fraction a/b. Rational

numbers include integers

Scientific Notation

A method of writing very large or very small numbers by using a number written

between 1 and 10 multiplied by a power of 10. A number written as the product of

two factors in the form , where n is an integer and

Standard Form of a

Number

Standard form is a way of writing down very large or very small numbers easily

Zero Exponent For every non-zero number a, a0 = 1.


Page 3

Exponential Form and Properties of Exponents

Vocabulary Labeled Example

Base

Exponent

Write each of these expressions in exponential form.

a. (−6) ∙ (−6) ∙ (−6) ∙ (−6) ∙ (−6) b. 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥

c. 8888888888

1

d.

77777

44444444


Page 4

Zero Property of Exponents and Negative Exponents

The exponent pattern: 24 = 2•2•2•2 = _______

23 = 2•2•2 = _______

22 = 2•2 = _______

21 = _______

20 = _____________ = ________

2-1 = ____________ = ________

2-2 = ____________ = ________

What happens to the product when you increase the exponent by one?

What happens to the product when you decrease the exponent by one?

Predict the answer for 20, 2-1, and 2-2. Be sure to follow your “rule!” When finished, discuss

your ideas with a partner.

Let’s try a different base: 54 = 5•5•5•5 = _______

53 = 5•5•5 = _______

52 = 5•5 = _______

51 = _______

50 = _______

5-1 = _______

5-2 = _______

5-3 = _______


Page 5

Zero Property of Exponents and Negative Exponents

For every nonzero number x, x0 = ____

For every nonzero number x, x-a = _____ Examples. Simplify each expression completely.

1. 40 2. (-7.8)0 3. b-5

4. 6-3 5. 9

1

b 6.

2

1

3

7. 3

6

a

m 8.

3

6

a

m

9.

6 3

1

m a

Zero and Negative Exponents

Simplify each expression

1) 4x0•50

2) 3-2

3) 2-4•2-8

4) (1

5)-3

5) -3a-2b-4 6) 5x0p-3

Rules


Page 6

0 0 2 4

6

-3

2 0

1) Simplify:

a) 6 b) 4 c) 5

2 1 d) e)

3 4

2) Evaluate when 2, 1, 3

a) 4

x y x

x

a b c

a b

-3

23 -2

-1

b) 5a

6c c) d) 2 b

ba

3) Simplify an • a-n. What is the mathematical relationship of

an and a-n? Justify your answer.

4) Are 3x-2 and 3x2 reciprocals? Explain.

5) Choose a fraction to use as a value for the variable a. Find the

values of a-1, a2, and a-2.


Page 7

Multiplication Property of Exponents

Ex. 4 56 6 = _______________________ =

6

Ex. 5e e = _______________________ =

e

Ex. 6 7 125f f f = _______________________ =

f

To multiply numbers or variables that are raised to a power, __________ the

exponents of the numbers or variables with the __________________________.

Examples. Simplify each expression completely.

1. 3 33 2. m5 m7 m87 3. a5

a b2 a11

4. x2y4x3y 5. (32)(3)(23) 6. c4 d7 c17

What do you do when there are coefficients?

Example: 6a33a 2a5

1. 6y2 3y3 2y4

2. 2y3 7x2 2y4

3. 5m 2p4 3m8

Rule


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Multiplication with Exponents

2 3 ?

?

5 5 (5 5) (5 5 5) 5

Rule: a bx x x

6 7

Example 1: Rewrite using one base.

4 4

4 2

3 5

-2 4 6 3

2 3 5 7

Example 2: Simplify

a)

b) 4 3

c) a

d) 4 3 6

a a a

x x

b a b

x y x y


Page 9

Let’s Try!

Simplify:

2 4

4 5

2 3

3 7 4

1)

2) 2 3

3) (6 )(2 )

4) (5 )(6 )

a a

x x

x y xy

x y x y

Find the area of each figure below. Write your answer is simplest

exponential form.

43x 39x

43x

26x


Page 10

Division Property of Exponents

Ex.

5

3

6 6 6 6 6 6 6

6 6 6 6 1

Ex.

4

2

x x x x xx

x x x

To divide numbers or variables with the same non-zero base, ____________ the

exponents. Or, look for where the base is “heavier” and leave the remainder.


1. 7

3

10

10 2. 25

18

x

x

3. 24

27

5

5 4. 4

4

3

3

5. 512m

3m 6. 36b

18

7. 3

2

5

2

8. 5 6

8 3

x y

x y

9. 3

2

3

10. 2

4

5

Rule


Page 11

Exponents: Powers of a Power

You can use what you learned about multiplying numbers with powers to find a shortcut for

simplifying expressions with powers. Complete each statement by showing equivalent

expressions. Let your final answer be written as a base raised to a single power (exponential

form).

1.) (36)2 = 36 36 = _____ 2. (54)3 = 54 54 54 =

3.) (27)4 = ______ • ______ • ______ • ______ = 4. (45)5 = ______ • ______ • ______ • ______ • ______ =

5.) (14)6 = 6.) (62)4 =

Look at your answers.

What do you notice about the two exponents in

the original expression as compared to the value

of the exponent in the final expression?

What operation would allow you to go straight

from the original two exponents to the final one?

To simplify a power to a power, _________________________ the exponents.


1. (22)3 2. (c5)4

3. (3●a)2 4. -52

5. (-5)2 6. (-4)2 ● -32

Rule


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Page 13


Page 14


Page 15


Page 16

Scientific Notation Notes (with powerpoint)

How wide is our universe?

210,000,000,000,000,000,000,000 miles

(_____ zeros) This number is written in _______________ ______________.

When numbers get this large, it is easier to write them in ____________ ___________.

A number is expressed in scientific notation when it is in the form

a x 10n

where ____ is between 1 and 10

and ___ is an integer

Write the width of the universe in scientific notation:

210,000,000,000,000,000,000,000 miles Where is the decimal point now?

Where would you put the decimal to make this number be between 1 and 10?

How many decimal places did you move the decimal?

When the original number is more than 1, the exponent is _________________.

The answer in scientific notation is:

Express 0.0000000902 in scientific notation.

Where would the decimal go to make the number be between 1 and 10?

The decimal was moved how many places?

When the original number is less than 1, the exponent is ____________________.

1) Write 28750.9 in scientific notation.

(choose one)

1. 2.87509 x 10-5

2. 2.87509 x 10-4

3. 2.87509 x 104

4. 2.87509 x 105


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2) Express 1.8 x 10-4 in standard notation.

3) Express 4.58 x 106 in standard notation.

Determine whether each of the following numbers is written in scientific notation? Explain.

Write each number in scientific notation.

4) 62,400

5) 0.00085

6) 1,602,000

Write each number in standard notation.

What does Scientific Notation look like on a calculator?

•Enter any 8-digit number into your calculator.

•Next, multiply by a 4-digit number.

•What do you see?


Page 18

Scientific Notation Homework

1. Is each number in scientific notation? If not, put the number in scientific notation.

1.6 x 10-6

950 x 105

0.84 x 10-5

8 x 103

2. Write 75,000,000,000 in scientific notation.

3. Write 0.0000429 in scientific notation.

4. Express 2.45 x 105 in standard form.

5. How much larger is 6 x 105 compared to 2 x 103

6. Which is the larger value: 2 x 106 or 9 x 105?


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7. A sample of bacteria triples every month. The expression 300 x 3m models a population of 300 bacteria

after m months of growth. Evaluate the expression for m = 0, 3, -2 and describe what each value of the

expression represents in the situation.

8. Recently, scientists have discovered 2 new moons. One moon’s distance from the sun is 234,000,000 miles,

while the other moon is 345,000,000 miles from the sun.

a. Write each number in scientific notation.

b. How many times closer to the sun is the first moon than the second moon? Write your answer in scientific

notation.

9. A person’s heart beats about 35 million beats in a year. If there are about 530 thousand minutes in a year,

what is the average heart rate in beats per minute?

10. The populations for four states are given below. List the states in order of their populations from least to

greatest.

Alaska: 6.19 x 105

Connecticut: 3.28 x 106

Hawaii: 1.18 x 106

North Carolina: 7.65 x 106


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Operations with Scientific Notation


Page 21

Multiplying with Scientific Notation

Example 1:

Example 2:

*The answer then must be changed to scientific

notation.

Example 3:

Example 4:

Dividing with Scientific Notation

Example 1:

*The answer then must be changed to scientific

notation.

Example 2:

Example 3:

Example 4:

How do the numbers compare to one another?

A. 4.5 106 B. 9 106 C. 4.5 107 How does B compare to A? How does C compare to A? How does A compare to C?


Page 22


Page 23

REAL WORLD APPLICATION

1) a. You are supposed to go to Idaho. It is 50 miles from here to Ogden. Then it is 90 miles to Pocatello

Idaho from Ogden. How far must you go?

b. You are supposed to go Venus. The earth is 9.3 x 107

miles from the sun. Venus is

8.5 x 1012 miles from the sun. How far is it to Venus?

2) a. You can travel 70 miles in one hour. How many hours will it take to get to Pocatello from Salt Lake

City?

b. You can travel 5.88 x 1012

miles in one light year. How many years will it take you to get to Venus?

3) a. The teeth of a comb are 3 millimeters wide. There are 45 teeth. How long is the comb?

b. A centipede’s leg is 7.23 x 10-2

cm. There are 50 legs on a side. How long is the centipede?

4) a. A bracelet weighs 8 oz. How many bracelets are in box which weighs a pound?

b. A grasshopper weighs 5.88 X 10-2

ounces. How many grasshoppers are in a pound? (a pound has 16

ounces)

5) Some stars in the Milky Way are 8 x 104 light years away.

Write this number in standard (expanded) form.

Why might scientists prefer to use this number in scientific notation?

6) A light year is 5.88 x 1012 miles.

Write this number in standard form.


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7) How many miles is it to the stars in the Milky Way: You’ll need the information in questions 1 and 2 to

answer this question. Show your work, including what you make the calculator do.

Write your answer in scientific notation.

8) If one eyelash measures 1.19 x 10-2 cm in diameter, and if your eyelashes lined up side by side in your eyelid

which measures 3 cm, how many eyelashes could fit on one eyelid?

Write your answer in standard form.

Write your answer in scientific notation.

If you lose 5 eyelashes per day, per eye, what percent of your total eyelashes are you losing per day?

9) A house spider weighs 4.22 x 10-3 ounces. How many house spiders are there in a pound? Note: there are

16 oz. in one pound. Show your work, including what you make the calculator do.


Page 25

Unit 13 Study Guide

EXPONENTS REVIEW

1. Any number to the power of zero always equals _________ because ____________________________

___________________________________________________________________________________.

2. If a number has a negative exponent, just __________________________________________________

___________________________________________________________________________________.

3. If two numbers with the same base are multiplying, just __________ the exponents.

4. If two numbers with the same base are dividing, just __________ the exponents.

5. If an exponent is beside a set of parentheses, just __________ it with the exponents inside the

parentheses.

6. If a negative sign is in front of parentheses that have an exponent outside, where does it fall in the

order of operations?

7. If a negative sign is inside parentheses that have an exponent outside, where does it fall in the order of

operations?

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 8.

a. –1 b. 0 c. –8.6 d. 1

____ 9.

a. b.

c.

d.

____ 10.

a.

b. 16 c.

d. –8

____ 11.

a.

b.

c.

d.


Page 26

____ 12.

a.

b.

c. d.

____ 13.

a. b. c. d.

____ 14.

a. b. c.

d.

____ 15.

a.

b. c.

d.

____ 16.

a. b. c. d.

____ 17.

a.

b.

c.

d.

SIMPLIFY. Pay attention to what you wrote above in the exponents review!

18. 72 75 19. a2b a3b4 20. nm

nm2

34

21. −(r5s4)0 22. 21

25

yx

yx

23. −2(2𝑥−3𝑦−1

4𝑥2𝑦−3)3

exponent


Page 27

SCIENTIFIC and STANDARD NOTATION—write the number in its equivalent other form.

24. 8,030,000,000 = ____________________ 25. 8.6 x 10-7 = ______________________

26. 8.72 x 105 = _____________________ 27. 0.0000073 = _____________________

Put in order from least to greatest.

28. 72 x 105, 6.9 x 106, 23 x 105 29. 19 x 10-3, 2.5 x 10-4, 1.89 x 10-4

Solve. Express your result in scientific notation.

30. 9.7821 x 10-17 + 3.14 x 10-18 31. 1.824 x 104 – 3.821 x 102

32. (1.5 x 105)(4 x 109) 32. (5.1 x 103)(1.63 x 10-5)

____ 33. Which number is written in scientific notation?

a. b. c. d.

____ 34. Which number is NOT written in scientific notation?

a. b. c. d.

____ 35. 0.0805

a. b. c. d.


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____ 36.

a. 9,000 b. c. 90,000 d. 360

____ 37. Order from least to greatest.

a. c. b. d.

____ 38. Which list shows the numbers in order from least to greatest?

a. c.

b. d.

____ 39.

a. b. c. d.

____ 40.

a. b. c. d.

____ 41. The diameter of Mercury is about miles. The diameter of Jupiter, the largest planet, is about

miles. What is the difference between the diameters of these planets expressed in scientific notation?

a. miles c. miles

b. miles d. miles

____ 42. The masses of four objects were measured during a physics experiment. The first and the last objects each

had a mass of 41.918 g. The second and the third objects each had a mass of 24.83 g. Find the total mass of the four

objects. Write your answer in scientific notation.

a. g c. g

b. g d. g

6.3 1064

6.3 1017

1.6 1064

1.6 1017

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Exponentsjtownsendmath.weebly.com/uploads/8/1/8/3/81834070/...You can use what you learned about multiplying numbers with powers to find a shortcut for simplifying expressions with