LEARN ▸ Scientific Notation, Exponents, and Irrational Numbers
What does this painting have to do with math?
Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
On the cover
Pan North IV, 1985
Al Held, American, 1928–2005
Acrylic on canvas
Private collection
Al Held (1928–2005), Pan North IV, 1985, acrylic on canvas, 72 x 84 in, private collection. © 2020 Al Held Foundation, Inc./Licensed by Artists Rights Society (ARS), New York
Great Minds® is the creator of Eureka Math® , Wit & Wisdom® , Alexandria Plan™, and PhD Science®
Published by Great Minds PBC. greatminds.org
© 2025 Great Minds PBC. All rights reserved. No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying or information storage and retrieval systems—without written permission from the copyright holder.
Printed in the USA
Module
1 Scientific Notation, Exponents, and Irrational Numbers
2 Rigid Motions and Congruent Figures
3 Dilations and Similar Figures
4 Linear Equations in One and Two Variables
5 Functions and Bivariate Statistics
6 Systems of Linear Equations
Student Edition: Grade 8, Module 1, Contents
Scientific Notation, Exponents, and Irrational Numbers
Topic
A
Introduction to Scientific Notation
Lesson 1
Large and Small Positive Numbers
Lesson 2
Comparing Large Numbers
Lesson 3
Time to Be More Precise—Scientific Notation
Lesson 4
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
Topic B
Properties and Definitions of Exponents
5
Topic C
Applications of the Properties and Definitions of Exponents
Lesson 11 .
Small Positive Numbers in Scientific Notation
159
161
23
41
59
75
Lesson 5 77
Products of Exponential Expressions with Whole-Number Exponents
Lesson 6
More Properties of Exponents
Lesson 7
Making Sense of the Exponent of 0
Lesson 8
Making Sense of Integer Exponents
Lesson 9
Writing Equivalent Expressions
Lesson 10
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
119
Lesson 12
Operations with Numbers in Scientific Notation
Lesson 13
Applications with Numbers in Scientific Notation
Lesson 14
Choosing Units of Measurement
Lesson 15
Get to the Point
Topic D
Perfect Squares, Perfect Cubes, and the Pythagorean Theorem
Lesson 16
Perfect Squares and Perfect Cubes
Lesson 17
Solving Equations with Squares and Cubes
Lesson 18
The Pythagorean Theorem
Lesson 19
Using the Pythagorean Theorem
Lesson 20
Square Roots
Topic E
Irrational Numbers
Lesson 21
Approximating Values of Roots and π 2
Lesson 22
Familiar and Not So Familiar Numbers
Lesson 23
Ordering Irrational Numbers
Lesson 24
Revisiting Equations with Squares and Cubes
Resources
Fluency Resources
Lesson 21
Sprint: Apply Properties of Positive Exponents
Sprint: Apply Properties of Positive and Negative Exponents
Sprint: Numerical Expressions with Exponents
Sprint: Scientific Notation and Negative Exponents
Sprint: Scientific Notation and Positive Exponents
Sprint: Squares
Sprint: Write
Student Edition: Grade 8, Module 1, Topic A
Million, Billion—What’s the Difference, Really?
Thousand
About 17 minutes
Seconds
Million Billion
About 32 years (103)(106) (109)
Feet
About 12 days
The rough length of a New York block The rough distance from New York to Boston
Dollars
About $0.03 every day of your life
People
A mid-sized high school
About $35 every day of your life
San Jose, CA (approximately)
The rough distance from New York to the moon
About $35,000 every day of your life
The rough population of North and South America combined
Million and billion sound awfully similar. Just one letter is different. Written out as powers of 10, they look similar too: it’s 10 6 vs. 10 9 . How different can they really be?
Extremely different, it turns out.
If a million seconds is a long vacation, then a billion seconds is longer than your life so far. If a million feet is a short flight in an airplane, then a billion feet is a journey to the moon. If a million dollars over your lifetime is a nice daily allowance, then a billion dollars over your lifetime is enough to buy a new car every day. If a million people is the population of a big city, then a billion people is the population of the whole Western Hemisphere.
Exponents let us write huge numbers by using just a few symbols. But don’t let that fool you into forgetting how huge the numbers might be and how different they are from one another.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
Large and Small Positive Numbers
Standard Form of a Number
1. Write each number in standard form.
Writing Very Large and Very Small Positive Numbers
2. Complete the table. The table shows the approximate measurements of objects seen in the demonstration.
Approximate Measurement (meters)
Form Unit Form Single Digit Times a Power of 10 (expanded form)
Single Digit Times a Power of 10 (exponential form)
Eiffel Tower (height)
3. Complete the table. The table shows the approximate measurements of objects seen in the demonstration.
Approximating Very Large and Very Small Positive Quantities
4. The length of Rhode Island, from the northernmost point to the southernmost point, is 77,249 meters.
a. Approximate the length of Rhode Island by rounding to the nearest ten thousand meters.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
5. The width of a smartphone is 0.0710 meters.
a. Approximate the width of the smartphone by rounding to the nearest hundredth of a meter.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
6. The length of California, from the northernmost point to the southernmost point, is 1,253,679 meters.
a. Approximate the length of California by rounding to the nearest million meters.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
7. The diameter of a water molecule is 0.000 000 000 28 meters.
a. Approximate the diameter of a water molecule by rounding to the nearest ten billionth of a meter.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
8. There are 3.2 ten thousand ounces in 1 ton.
a. Approximate the number of ounces in 1 ton by rounding to the nearest ten thousand ounces.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
1. Consider the number 7,123,456
a. Approximate the number by rounding to the nearest million.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
2. Consider the number 0.000 0285
a. Approximate the number by rounding to the nearest hundred thousandth.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
Large and Small Positive Numbers
In this lesson, we
• explored large and small positive numbers by relating them to the size of real-world objects.
• analyzed equivalent forms of large and small positive numbers.
• approximated very large and very small positive numbers.
Examples
1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the Eiffel Tower.
Approximate Number of Stacked Pennies
Single Digit Times a Power of 10 (exponential form) Eiffel Tower 200,000 2 hundred thousand 2 × 100,000 2 × 10 5
Standard Form Unit Form
Single Digit Times a Power of 10 (expanded form)
Use place value units when writing numbers in unit form. The expanded form of one hundred thousand is 100,000. The exponential form of one hundred thousand is 10 5 .
2. Complete the table. The table shows the average speed in miles per hour of a starfish.
Each number in the Starfish row is an equivalent form of 0.01
3. There are 4,356,000 square feet in 100 acres.
A unit fraction has a numerator of 1
a. Approximate the number of square feet in 100 acres by rounding to the nearest million square feet.
4,000,000 square feet
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
4 × 10 6 square feet
Writing equivalent forms of 4,000,000 may help when writing the number as a single digit times a power of 10 in exponential form.
4,000,000 = 4 million = 4 × 1,000,000 = 4 × 10 6
Student Edition: Grade 8, Module 1, Topic A, Lesson 1
Name Date
For problems 1–5, write the number in standard form.
trillion
6. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of a given object.
Approximate Number of Stacked Pennies
Standard Form Unit Form
Single Digit Times a Power of 10 (expanded form)
Single Digit Times a Power of 10 (exponential form)
7. Complete the table. The table shows the average speed of a given animal.
Average Speed (miles per hour)
Standard Form Unit Form Fraction
Single Digit Times a Unit Fraction (expanded form)
Single Digit Times a Unit Fraction (exponential form)
8. The deepest part of the ocean, called Challenger Deep, is 36,200 feet below sea level.
a. Approximate the depth of Challenger Deep by rounding to the nearest ten thousand feet.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
9. The smallest insect on the planet is a type of parasitic wasp and measures 0.000 139 meters.
a. Approximate the length of the insect by rounding to the nearest ten thousandth of a meter.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
10. There are 907,200,000 milligrams in 1 ton.
a. Approximate the number of milligrams in 1 ton by rounding to the nearest hundred million milligrams.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
11. There are 0.000 000 001 102 293 tons in 1 milligram.
a. Approximate the number of tons in 1 milligram by rounding to the nearest billionth of a ton.
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
12. The world population is expected to reach 9.7 billion people in the year 2050.
a. Approximate the expected world population in 2050 by rounding to the nearest billion people.
b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.
Remember
For problems 13–16, add or subtract.
For problems 17–19, use the distributive property to write an equivalent expression.
20. Match each expression to the correct power of 10 in exponential form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
Comparing Large Numbers
1. Analysts estimate that there were about 9 billion devices worldwide that used wireless routers to wirelessly connect to the internet in 2017 and 2018. A wireless router supports about 300 devices.
The number of devices worldwide is about how many times as much as the number of devices one router can support?
Unknown Factor
For problems 2–4, write each number as a single digit times a power of 10 in exponential form and write an unknown factor equation. Write your answer as a single digit or as a single digit times a power of 10 in exponential form.
2. 9 billion is how many times as much as 3000?
3. 9 million is how many times as much as 3,000,000?
4. What number is 3000 times as much as 30?
Times As Much As
For problems 5–8, approximate each quantity as a single digit times a power of 10 in exponential form and write the unknown factor equation. Then solve for the unknown factor by using a strategy of your choice.
5. In the fall of 2019, approximately 50,800,000 students attended school in the United States in prekindergarten through grade 12. About 11,683,000 of those students attended middle school.
The overall number of students in prekindergarten through grade 12 was about how many times as much as the number of students in middle school?
6. In 2019, the total outstanding consumer debt in the United States, including mortgages, auto loans, credit cards, and student loans, was about $3.9 trillion. That same year, the published national debt of the United States was $22,460,468,000,000
The published United States national debt was about how many times as much as the total United States consumer debt?
7. As of July 1, 2018, the US Census Bureau estimated the population of New York City at 8,398,748 people and the population of New York State at 19,542,209 people.
The population of New York State was about how many times as much as the population of New York City?
8. The total global carbon dioxide emissions for 2018 was about 33.1 billion tons. That same year, the carbon dioxide emissions by natural gas in the United States was about 1.629 billion tons.
Approximately what fraction of the total global carbon dioxide emissions was from the use of natural gas in the United States?
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
Bacterial life appeared on Earth about 4 billion years ago. Insects appeared about 400,000,000 years ago.
a. Write when bacterial life appeared on Earth as a single digit times a power of 10 in exponential form.
b. The number of years since bacterial life appeared on Earth is about how many times as much as the number of years since insects appeared?
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
Comparing Large Numbers
In this lesson, we
• wrote unknown factor equations to answer how many times as much as questions.
• approximated large numbers by writing them as a single digit times a power of 10.
• wrote out factors of 10 to help evaluate quotients and find unknown factors.
Examples
For problems 1 and 2, write each quantity as a single digit times a power of 10 in exponential form, and write an unknown factor equation. Then find the answer to the question.
1. 6,000,000 is how many times as much as 3000?
6,000,000 = 6 × 10 6 3000 = 3 × 10 3
= ⋅ 3 × 10 3
Writing out the factors of 10 makes the division simpler because 10 10 is 1.
= 2000
6,000,000 is 2000 times as much as 3000.
2. 90,000 is how many times as much as 4000? 90,000 = 9 × 10 4 4000 = 4 × 10 3 9 × 10 4 = ⋅ 4 × 10 3
The question can be reworded as 4000 times what number is 90,000?
90,000 is 22 1 2 times as much as 4000
3. The land area of France is 212,954 square miles. The land area of Canada is 3,854,083 square miles. The land area of Canada is about how many times as large as the land area of France? 212,954 ≈ 200,000 = 2 × 10 5 3,854,083 ≈ 4,000,000 =
Ask, “Which quantity is being multiplied?” to decide how to write the unknown factor equation.
Approximate the value of each land area. Then write the approximation as a single digit times a power of 10 in exponential form.
= 2 × 10 = 20
The land area of Canada is about 20 times as large as the land area of France.
Student Edition: Grade 8, Module 1, Topic A, Lesson 2
Name Date
PRACTICE
For problems 1–3, compute mentally.
1. 600 is how many times as much as 200?
2. 600 is how many times as much as 20?
3. 600 is how many times as much as 2?
For problems 4–6, write an unknown factor equation to represent the question. Then find the answer to the question.
4. 800,000 is how many times as much as 2000?
5. 60,000,000 is how many times as much as 30,000?
6. 6 × 10 5 is 2 × 10 3 times as much as what number?
For problems 7–11, use the values in the table to answer the questions.
Approximate
7. Based on land area, about how many islands the size of Jamaica does it take to equal the size of Russia?
8. The land area of the United States is about how many times as large as the land area of Brazil?
9. The land area of Belize is about how many times as large as the land area of Brazil?
10. Which country’s land area is about 1000 times as large as the land area of Jamaica?
11. Which country’s land area is about 1 500 as large as the land area of the United States?
12. The Atlantic Ocean contains about 310,410,900 cubic kilometers of water. Lake Superior, which is the largest lake in the United States, contains about 12,000 cubic kilometers of water.
Approximately how many Lake Superiors would it take to fill the Atlantic Ocean?
Remember
For problems 13–16, add or subtract.
17. Approximate 8,538,206 by rounding to the nearest million. Write your approximation as a single digit times a power of 10 in exponential form.
18. Which of the following are equivalent to 5.206? Choose all that apply.
A. 5 ones 2 tenths 6 thousandths
B. 5 + 0.2 + 0.06
C. (5 × 1) + (2 × 1 10 ) + (6 × 1 100 )
D. (5 × 1) + (2 × 0.1) + (6 × 0.001)
E. 5206 thousands
F. 5206 1000
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
Time to Be More Precise—Scientific Notation
Archimedes was a Greek mathematician whose ideas were ahead of his time. He lived in Sicily during the third century BCE and was the first to develop fundamental concepts in geometry, calculus, and physics.
Fascinated by very large numbers and living on the coast of the Ionian Sea, Archimedes set out to determine how many grains of sand are needed to fill the known universe.
Mathematicians used Ionic Greek notation in the third century BCE. So instead of numbers, they used symbols and letters from the Greek alphabet.
3. Read the story on the death of Archimedes. Then write the Greek math symbols as numbers in the blanks that follow each number. Because the exact details of his death are not confirmed, this story is one of many theories about how Archimedes died during the Second Punic War.
Archimedes’s native city of Syracuse, Italy, was captured by Roman forces in the year σιβ BCE ( BCE). While the city was under siege, Archimedes was drawing a diagram of circles in the sand.
As he contemplated his work, the οε-year-old ( -year-old) mathematician was approached by a Roman soldier demanding that Archimedes meet immediately with the general of the Roman Army. Archimedes refused, insisting that he would not go until he finished his math problem. This infuriated the Roman soldier, and he pulled out his sword and killed Archimedes on the spot.
Archimedes’s last words are thought to be “Do not disturb my circles.” His work would go unknown until the year φλ CE, ψμβ years ( CE, years) after his death.
Another Way to Represent Numbers
4. Fill in the blanks to complete the statement.
A number is written in scientific notation when it is represented as a number a multiplied by a of .
The general expression that represents a number written in scientific notation is × .
The absolute value of a must be at but than .
5. Identify the first factor and the order of magnitude of the expression 8.86 × 10 6 .
6. Use the definition of scientific notation to indicate whether each number is an example or a nonexample.
× 10 2
× 10 2
× 10 6
× 10 3
× 10 7 −6.75 × 10 6 1 × 10 9 10 × 10 10
Interpreting Scientific Notation
For problems 7–9, write the number in standard form.
3 × 10 2
10. Use the Place Value Chart to write the number 9.1 × 10 3 in standard form.
11. Use the Place Value Chart to write 2.05 × 10 6 in standard form.
12. Logan writes the number 6.7 × 10 3 in standard form. He writes 67,000 because 10 3 represents thousands. Do you agree with Logan? Explain.
Using Scientific Notation
13. Write 200,000 in scientific notation.
14. Use the Place Value Chart to write 350 in scientific notation.
For problems 15–18, write the number in scientific notation. Use the Place Value Chart, if needed.
15. 47,500,000
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
1. In 2018, the population of Italy was approximately 60,630,000 people. Write this number in scientific notation.
2. The average human body contains about 3.4 × 10 10 cells. Write this number in standard form.
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
Time to Be More Precise—Scientific Notation
In this lesson, we
• learned the definition of scientific notation.
• identified examples and nonexamples of numbers written in scientific notation.
• wrote numbers in scientific notation and in standard form.
Examples
1. Circle all the values written in scientific notation. 9.82 × 10 15 million 0.6 × 10 11 −6 × 10 5 4.2 × 10 3 4,200,000
Terminology
• A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10
• The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation.
Numbers written in scientific notation must be in the form a × 10 n
The absolute value of a, the first factor, must be at least 1 but less than 10.
2. Write the number 3.08 × 10 4 in standard form.
3.08 × 10 4 = 30,800
The order of magnitude, 4, shows that the highest place value of the number written in standard form is the ten thousands place.
Consider using a place value chart.
The order of magnitude is 6
3. A number is represented on a place value chart. Write the number in scientific notation. 10 6 10 5 10 4 10 3 10 2 10 1 1 7 5 0 0 0 0 0
7,500,000 = 7.5 × 10 6
The first factor is 7.5.
4. The earliest known mammal on Earth is a tiny mouse-like creature called the morganucodontid. It lived about 210,000,000 years ago. Write this number in scientific notation.
210,000,000 = 2.1 × 10 8
The first factor must have an absolute value of at least 1 but less than 10, so the first factor is 2.1
The highest place value is hundred millions, so the order of magnitude is 8
Student Edition: Grade 8, Module 1, Topic A, Lesson 3
Name Date
PRACTICE
1. Circle all the numbers written in scientific notation.
× 10 7
For problems 2–7, write the number in standard form.
1 × 10 7
3.3 × 10 5
5.505 × 10 3
× 10 2
8. In 2017, about 8.3 × 10 12 text messages were sent and received worldwide. Write this number in standard form.
9. In 2014, the United States discarded a total of 5.08 × 10 9 pounds of trash. Write this number in standard form.
10. Sara believes that the number 3 × 10 is not written in scientific notation. Do you agree or disagree? Explain.
11. Write the number shown on the place value chart in scientific notation.
For problems 12–16, write the number in scientific notation.
17. In 2019, the man considered to be the richest person in the world had a net worth totaling about $111 billion. Write this number in scientific notation.
18. Match each number written in standard form with its corresponding number written in scientific notation.
19. The table represents the box office sales of the highest grossing movie in 2019 in various markets. Write each number in standard form and in scientific notation.
Remember
For problems 20–23, add or subtract.
20. 1 10 + 1 5 21. 1 4 + 1 8
1 3 1 9
24. Consider the number 0.000 236.
a. Approximate the number by rounding to the nearest ten thousandth.
1 5 1 15
b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.
For problems 25–30, add or subtract.
7 − 4 26. −7 + (−4) 27. −7 + 4
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
For problems 1–6, find the sum.
1. 3 thousands + 2 thousands + 4 thousands
Adding and Subtracting: What Is the Same?
7. Pedro and Ava each find the sum 3 × 10 5 + 2 × 10 5 + 4 × 10 5 . Pedro uses the order of operations, and Ava uses the distributive property.
Pedro’s work:
Ava’s work:
a. Who is correct? Explain.
b. Compare the two methods.
For problems 8–10, add or subtract. Write the answer in scientific notation.
8. 3 × 10 12 + 2 × 10 12 + 4 × 10 12
9. 3 × 10 8 + 2 × 10 8 − 4 × 10 8
3.6 × 10 9 − 2.1 × 10 9 + 4.4 × 10 9
11. The table shows the number of views for three popular online videos.
a. How many total views do the three videos receive? Write the answer in scientific notation.
b. How many more views does the video of the cat singing receive than the video of the baby dancing? Write the answer in scientific notation.
Rewriting Sums and Differences
12. Find the sum 6 × 10 3 + 7 × 10 3 + 8 × 10 3 .
For problems 13 and 14, add or subtract. Write the answer in scientific notation.
9.25 × 10 5 + 9.8 × 10 5
14. 5 × 10 6 + 9 × 10 6 − 2 × 10 6 + 8 × 10 6
15. In 2008, a bakery in Indonesia set a record for creating the world’s tallest cake, which was 108.27 feet tall. Bakers used about 62 thousand ounces of powdered sugar and about 5.7 × 10 4 ounces of margarine in the cake.
How many total ounces of powdered sugar and margarine were in the cake? Write the answer in scientific notation.
Adding and Subtracting: What Is Different?
For problems 16 and 17, add or subtract. Write the answer in scientific notation.
16. 4 × 10 5 + 3 × 10 6
7.2 × 10 5 − 4 × 10 4
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
The table shows the estimated number of stars in four different galaxies.
Galaxy Estimated Number of Stars
Milky Way
Whirlpool
2.5 × 10 11
1.0 × 10 11 Sunflower
Antennae
4.0 × 10 11
3.0 × 10 11
What is the estimated total number of stars in the four galaxies? Write the answer in scientific notation.
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
Adding and Subtracting Numbers Written in Scientific Notation (Optional)
In this lesson, we
• added and subtracted numbers written in scientific notation.
• rewrote sums and differences in scientific notation.
Examples
For problems 1–3, add or subtract. Write the answer in scientific notation.
Apply the distributive property because all three terms have the same power of 10
The first factor, 17, is greater than 1 but not less than 10 Rewrite 17 as 1.7 × 10.
3. 4 × 10 6 + 9 × 10 5
4 × 10 6 + 9 × 10 5 = 4 × 10 × 10 × 10 × 10 × 10 × 10 + 9 × 10 × 10 × 10 × 10 × 10 = (4 × 10) × 10 × 10 × 10 × 10 × 10 + 9 × 10 × 10 × 10 × 10 × 10
To apply the distributive property, first rewrite 4 × 10 6 as 40 × 10 5 so the powers of 10 are the same.
= 40 × 10 5 + 9 × 10 5 = (40 + 9) × 10 5 = 49 × 10 5 = (4.9 × 10) × 10 5 = (4.9 × 10) × 10 × 10 × 10 × 10 × 10 = 4.9 × (10 × 10 × 10 × 10 × 10 × 10) = 4.9 × 10 6
Student Edition: Grade 8, Module 1, Topic A, Lesson 4
Name Date
For problems 1 and 2, find the sum.
1. 4 thousands + 3 thousands
For problems 3 and 4, add or subtract. Write the answer in scientific notation.
5. The table shows the estimated number of US households that had a pet in 2019.
a. What is the estimated total number of households that had a bird or a saltwater fish? Write the answer in scientific notation.
b. About how many more households had a reptile than had a saltwater fish? Write the answer in scientific notation.
For problems 6–10, add or subtract. Write the answer in scientific notation.
6. 9 × 10 8 − 2.7 × 10 8
11. The table shows an estimated number of smartphone users in the three most populous countries in the world in 2018.
Country
Estimated Number of Smartphone Users China 783,000,000 India
2.52 × 10 8
a. What is the estimated total number of smartphone users for the three countries in 2018? Write the answer in scientific notation.
b. In 2018, how many more smartphone users were estimated to be in China than in the United States and India combined? Write the answer in scientific notation.
12. Evaluate 5 × 10 5 + 3 × 10 4 . Write the answer in scientific notation.
13. Nora creates a website password with six characters.
• There are about 3.089 × 10 8 possible six-character passwords that use only lowercase letters.
• There are about 2.177 × 10 9 possible six-character passwords that use any combination of only lowercase letters, only numbers, or both lowercase letters and numbers.
How many passwords can Nora create that use at least one number?
Remember
For problems 14–17, add or subtract.
18. Write 9003 in scientific notation.
19. Which expressions are equivalent to 2 3 ( 4)? Choose all that apply.
Student Edition: Grade 8, Module 1, Topic B
Who Would Win?
The Eighth Power of 10
A Whole Squadron of Seventh Powers of 10
It’s easy to forget how different 10 7 and 10 8 really are from one another. They look like they’re just one apart. And, in a sense, they are: one factor of 10.
That means that 10 8 is a full 10 times as big as 10 7 . In the picture above, neither would win; they’re equivalent!
In this topic, we will explore different ways to combine powers. What happens if you multiply 10 8 and 10 7 ? What if you divide 10 8 and 10 7 ? These patterns will lead us to whole new frontiers where we will try to make sense of expressions like 10 0 and 10 −4
If you get lost, remember this: 10 8 is 10 times the size of 10 7 !
Student Edition: Grade 8, Module 1, Topic B, Lesson 5
Name
Products of Exponential Expressions with Whole-Number Exponents
Date
1. Multiply. Write the product as a power of 10 in exponential form.
Multiplying Powers with Like Bases
2. Multiply. Write the product as a power of 10 in exponential form.
3. Multiply. Write the product as a
Applying the Property of Exponents
For problems 4–14, apply the property of exponents to write an equivalent expression.
Multiplying Powers with Unlike Bases
For problems 16–22, apply the property of exponents to write an equivalent expression.
Student Edition: Grade 8, Module 1, Topic B, Lesson 5
Properties of Exponents
Description
Property
Example(s)
Description
Definitions of Exponents
Definition(s)
Example(s)
Student Edition: Grade 8, Module 1, Topic B, Lesson 5
Name Date
For problems 1–4, apply the property of exponents to write an equivalent expression. 1. 5 3 ⋅ 5 6
(−5) 3 (−5) 8
3. (3 5 )4 (3 5 )6
5. Abdul finds an equivalent expression for
Then he counts the 4’s to determine the exponent to use in his answer.
Explain a quicker way for Abdul to get the same result.
Student Edition: Grade 8, Module 1, Topic B, Lesson 5
Name Date
Products of Exponential Expressions with Whole-Number Exponents
In this lesson, we
• discovered a pattern when multiplying powers with like bases.
• learned the product of powers property.
• applied the property of exponents to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number m and n are positive whole numbers when x m x n = x m+n
Examples
For problems 1–3, apply the property of exponents to write an equivalent expression.
1. 9 4 ⋅ 9 2 9 4 ⋅ 9 2 = 9 4+2
94 is 4 factors of 9. 92 is 2 factors of 9. So 94 . 92 is 4 + 2 factors of 9, which can be written as 94+2. 94 .
times
2. (−6)(−6) 3 (−6)(−6) 3 = (−6) 1+3
Negative bases and fractional bases have parentheses to avoid confusion.
In this problem, there are two different bases: a base of 7 and a base of 8
First, group the powers with like bases together. Then, apply the property of exponents to each base separately.
Student Edition: Grade 8, Module 1, Topic B, Lesson 5
Name Date
PRACTICE
For problems 1–12, apply the property of exponents to write an equivalent expression.
13. Which of the following is equivalent to 5 3 ⋅ 5 5 ? Choose all that apply.
5 3+5
14. Which of the following is equivalent to 9 12 ? Choose all that apply.
15. Sara states that when you multiply two powers with the same base, the exponents are multiplied together. She uses an example to support this claim:
Fill in the boxes to create an equation that shows that Sara’s claim is incorrect.
16. Find the area of the rectangle. Write the area as a single base raised to an exponent.
105 ft
102 ft
For problems 17–20, indicate whether each result is a positive or a negative number.
(−3) 2
21. The product (−1) 3 ⋅ (−1) n is negative. Which of the following values of n are possible? Choose all that apply.
A. 2
B. 5
C. 7
D. 8
E. 10
For problems 22–27, solve for b.
28. Fill in the boxes with digits 1–6 to make each equation true. Each digit can be used only once.
Remember
For problems 29–32, add or subtract.
33. Evaluate 2.1 3 .
A. 4.41
B. 6.3
C. 8.001
D. 9.261
Student Edition: Grade 8, Module 1, Topic B, Lesson 6
Name Date
More Properties of Exponents
1. Write an expression equivalent to 34 ⋅ 34.
2. Write an expression equivalent to 34 ⋅ 34 ⋅ 34.
3.
Raising Powers to Powers
4. Without knowing what is under the mustard spot, what do we know about the expression?
)5
5. Draw mustard spots to create an expression on the right side of the equal sign to make a true statement. ( )5 =
6. Assume 34 is under the mustard spot.
( )5
a. Write factors of 34 to create an expression on the right side of the equal sign to make a true statement. (34)5 =
b. How is (34)5 similar to the mustard spot expression in problem 5?
c. Explain the meaning of (34)5.
d. Apply the properties of exponents to write an equivalent expression for (34)5. Explain your reasoning.
Raising Products to Powers
7. Assume 4 ⋅ 3 is under the mustard spot. In this case, what is repeated? How many times?
( )5
8. Vic notices that two problems he already completed are now also covered in mustard. What do you think the original expressions are?
9. Students are discussing possible answers to a problem. What do you think?
(4x)3 = (4x) (4x) (4x) (4x)3 = 43x 3 It can’t be both, but which one is right?
Raising Quotients to Powers
10. Consider the expression ( x y ) 4, where y ≠ 0. Use what you know about exponents and their properties to write an equivalent expression.
11. Assume 3 4 is under the mustard spot. What equivalent expression should Vic write?
Student Edition: Grade 8, Module 1, Topic B, Lesson 6
Name Date
For problems 1–3, apply the properties of exponents to write an equivalent expression. Assume y is nonzero.
1.
3 ) 6
x) 4
3. (3 4 y )2
Student Edition: Grade 8, Module 1, Topic B, Lesson 6
Name Date
More Properties of Exponents
In this lesson, we
• established two new properties of exponents.
• used the properties of exponents to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number
Power of a Power Property
m and n are positive whole numbers when x m x n = x m+n
x is any number m and n are positive whole numbers when (x m)n = x m n
Power of a Product Property
x and y are any numbers n is a positive whole number when (xy)n = x ny n
For problems 1–3, apply the properties of exponents to write an equivalent expression.
(w 2 ) 3 w 2 ⋅ 3
4 times 3. ( 9 2 10 )6 9 2 6 10 6 Both the numerator, 9 2 , and the denominator, 10, are raised to the sixth power.
(w2)3 = w2 w2 w2 = w2+2+2 = w3.2 3 times (w 2 ) 3 is 3 factors of w 2 . This is 3 factors of 2 factors of w, which is 3 ⋅ 2 factors of w. The entire term 3 r 7 is raised to the fourth power (3r7)4 = 3r
. 3
Student Edition: Grade 8, Module 1, Topic B, Lesson 6
Name Date
PRACTICE
For problems 1–10, apply the properties of exponents to write an equivalent expression. Assume m is nonzero. 1. (3 4 ) 5
(8 c) 9
(dg) 11
(3 8 4 6 )6
( n 3 m 7 )5
((p 2 ) 4 ) 6
11. Jonas says (5n 2 ) 3 is equivalent to 5n 2 3 . Do you agree? Explain your reasoning.
In problems 12–14, fill in the boxes with values that make the equation true. Assume s is nonzero.
12. (a 3 b 4 ) = a 6 b 13. (r s 3 )2 = r 10 s
( × 10 5 ) 2 = 9 × 10
15. Which expressions are equal to 8 24 ? Choose all that apply.
(8 20 ) 4
(8 12 ) 2
8 8 ⋅ 8 3
8 14 ⋅ 8 10
8 8 + 8 16
16. Which expressions are equal to x 12 y 8 z 4 ? Choose all that apply.
A. (x 8 y 6 z) 4
B. (x 3 y 2 z) 4
C. (x 6 y 4 z 2 ) 2
D. (x 3 y 4 z 2 )(x 4 y 2 z 2 )
E. (x 6 y 5 z)(x 6 y 3 z 3 )
17. The edge of a cube measures 4 n 2 inches. What is the volume of the cube?
18. The side length of a square measures 3 x 4 y 5 meters. What is the area of the square?
Assume y is nonzero.
Remember
For problems 19–22, add or subtract.
For problems 23–25, apply the properties of exponents to write an equivalent expression.
23. 7 3 ⋅ 7 5
25. (5 7 )8 (5 7 )2
26. Which expression is equivalent to −11 − 7?
A. −11 − (−7)
B. 11 + 7
C. −11 + (−7)
D. 11 − (−7)
24. (−8) ⋅ (−8) 6
Student Edition: Grade 8, Module 1, Topic B, Lesson 7
Name Date
Making Sense of the Exponent of 0
Defining the Exponent of 0
1. Could 10 0 = 10? Use the product 10 0 ⋅ 10 3 to show whether it upholds the property x m ⋅ x n = x m + n .
2. Could 10 0 = 0? Use the product 10 0 ⋅ 10 3 to show whether it upholds the property
x m ⋅ x n = x m + n
3. Could 10 0 = 1? Use the product 10 0 ⋅ 10 3 to show whether it upholds the property
x m x n = x m + n .
In problems 4 and 5, the expanded form of a number written with exponential notation is given. Write the number in standard form.
4. 4 × 10 2 + 3 × 10 1 + 6 × 10 0
5. 7 × 10 3 + 5 × 10 2 + 1 × 10 0
In problems 6 and 7, the standard form of a number is given. Write the number in expanded form by using exponential notation.
6. 982
7. 10,735
8. Apply the properties of exponents to write an equivalent expression for x 0 ⋅ x n . Then show that the definition x 0 = 1 upholds the property x m ⋅ x n = x m + n for any nonzero x.
Using the Exponent of 0
Determine the value of n in problems 9–12.
9. 11 4 ⋅ 11 0 = 11 n
10. (−4) 2 (−4) n = (−4) 2 11. 10 n ⋅ 10 3 = 1000 12. 5 0 ⋅ 5 0 = n
Student Edition: Grade 8, Module 1, Topic B, Lesson 7
Name Date
For problems 1–4, apply the properties of exponents and the definition of the exponent of 0 to write an equivalent expression.
Student Edition: Grade 8, Module 1, Topic B, Lesson 7
Name Date
Making Sense of the Exponent of 0
In this lesson, we
• made predictions about the value of a power with an exponent of 0.
• used properties of exponents to test which prediction was true about the value of a power with an exponent of 0.
• learned the definition of the exponent of 0.
Terminology
A power with an exponent of 0 is x 0 = 1 for any nonzero x
• applied the definition of the exponent of 0 to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number m and n are whole numbers when
Power of a Power Property
x is any number m and n are whole numbers when
Power of a Product Property
x and y are any numbers n is a whole number when (
Definition of the Exponent of 0 x is nonzero
Examples
For problems 1–4, apply the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
The exponent of 0 applies only to the base of c, not to the entire expression 5 c
Apply the definition of the exponent of 0 to write 10 0 as 1
Solve by using either strategy.
5. Solve for q. (4 5 )0 (4 5 )q = (4 5 )15 (4 5 )0 (4 5 )q = (4 5 )15 ( 1 ) (4 5 )q = (4 5 )15 (4 5 )q = (4 5 )15 q = 15 (4 5 )0 (4 5 )q = (4 5 )15 (4 5 )0 + q = (4 5 )15 (4 5 )q = (4 5 )15 q = 15
The solution is 15.
6. Find the value of 8 × 10 3 + 6 × 10 2 + 2 × 10 0 . 8 × 10 3 + 6 × 10 2 + 2 × 10 0 = 8 × 1000 + 6 × 100 + 2 × 1 = 8000 + 600 + 2 = 8602
Student Edition: Grade 8, Module 1, Topic B, Lesson 7
Name
For problems 1–6, apply the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
For problems 7–18, apply the properties of exponents and the definition of the exponent of 0 to write an equivalent expression. Assume all variables are nonzero.
For problems 19–23, apply the properties of exponents to determine the value of t.
19. 8 0 ⋅ 8 2 = 8 t 20. (−9) 3 (−9) t = (−9) 3
21. (3 8 )0 (3 8 )t = (3 8 )11
2 5 ⋅ 2 0 = t
23. (−0.25) 0 (−0.25) t = 1
24. Fill in the boxes with digits 0–5 to make each equation true. Each digit can be used only once.
a. (−2) 0 (−2) = (−2) 5 b. (1 4 )(1 4 )0 = 1
(6 0 ) 2 =
(m n ) 0 (m n ) = 1
25. Choose the expression with a value of 9804.
A. 9 × 10 3 + 8 × 10 2 + 4 × 10 1
B. 9 × 10 4 + 8 × 10 3 + 4 × 10 1
C. 9 × 10 4 + 8 × 10 3 + 4 × 10 0
D. 9 × 10 3 + 8 × 10 2 + 4 × 10 0
26. Find the value of 6 × 10 4 + 3 × 10 2 + 2 × 10 0 .
A. 632
B. 6032
C. 6320
D. 60,302
E. 60,300
27. So-hee states that 4 0 = 4. Do you agree? Explain your reasoning by using the properties of exponents.
Remember
For problems 28–31, add or subtract.
28. 1 6 + 1 4 29. 2 9 + 7 6
30. 5 6 1 8
11 10 1 25
For problems 32–35, apply the properties of exponents to write an equivalent expression.
(4 5 ) 6
36. Which expression is equivalent to 75? Choose all that apply.
3 2 ⋅ 5
3 ⋅ 5 ⋅ 5
E. 3 ⋅ 3 ⋅ 5
Student Edition: Grade 8, Module 1, Topic B, Lesson 8
Name Date
Making Sense of Integer Exponents
1. Use the properties of exponents to find the value of 10 4 ⋅ 10 −4 .
Integer Exponents
For problems 2 and 3, suppose the properties of exponents remain true for negative exponents.
2. Are 10 4 and 10 −4 multiplicative inverses? Why?
3. Write an equivalent expression for 10 −4 .
For problems 4–9, use the definition of negative exponents to write an equivalent expression. Assume that x is nonzero.
For problems 10 and 11, use the definition of negative exponents to write an equivalent expression with positive exponents.
10. 1 10 2 11. 1 4 6
Quotients of Powers
So-hee applies the properties of exponents to write an equivalent expression for 103 10 7 . So-hee’s work is shown.
12. Apply the properties of exponents and the definition of negative exponents to verify So-hee’s answer.
For problems 13–15, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
13. 15 3 ⋅ a 6 a 3
14. 7 d 4 14 d 7
Student Edition: Grade 8, Module 1, Topic B, Lesson 8
Name Date
1. Use the definition of negative exponents to write 1 5 7 with a negative exponent.
2. Use the definition of negative exponents to write 1 ____ g −3 with a positive exponent. Assume g is nonzero.
3. Kabir and Yu Yan each write an equivalent expression for 8 9 8 5 . The table shows their work.
Kabir’s Work Yu Yan’s Work
Explain which property of exponents and/or which definition of exponents that Kabir and Yu Yan use to get the final expression.
Student Edition: Grade 8, Module 1, Topic B, Lesson 8
Name Date
Making Sense of Integer Exponents
In this lesson, we
• related negative exponents to multiplicative inverses.
• learned the definition of a negative exponent.
• applied the definition of a negative exponent to write equivalent expressions.
Product of Powers with Like Bases Property
x is any number m and n are integers when
Power of a Power Property
x is any number m and n are integers when (
Power of a Product Property
x and y are any numbers n is an integer when (xy)n = x ny n
Definition of the Exponent of 0 x is nonzero x 0 = 1
Definition of Negative Exponents x is nonzero n is an integer when x –n = 1 x n
Examples
1. How can we show 10 −4 = 1 10 4 ?
By using the properties of exponents, we know 10 4 ⋅ 10 −4 = 10 4+(−4) = 10 0 = 1
We also know 10 4 ⋅ 1 10 4 = 1
This means 10 −4 and 1 10 4 are both multiplicative inverses of 10 4 . So 10 −4 = 1 10 4 . Two factors that have a product of 1 are multiplicative inverses.
For problems 2–4, use the definition of negative exponents to write an equivalent expression. Assume p is nonzero. 2. 17 −4 1 17 4
The multiplicative inverse of p 6 can be expressed as 1 p 6 or p −6 .
A fraction represents a division statement.
5. Apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume r is nonzero. 2 r 5 16 r 11
Solve by using either strategy.
Student Edition: Grade 8, Module 1, Topic B, Lesson 8
Name Date
For problems 1–9, use the definition of negative exponents to write an equivalent expression. Assume all variables are nonzero.
For problems 10–12, use the definition of negative exponents to write an equivalent expression.
For problems 13–19, apply the properties and definitions of exponents to write an equivalent expression with positive exponents. Assume all variables are nonzero.
20. Maya says 10 −5 is the same as (−10) 5 . Do you agree? Explain your reasoning.
21. Which one does not belong? Circle your answer and explain your reasoning.
22. Order the values from least to greatest.
Remember
For problems 23–26, add or subtract.
23. 5 6 + 3 4 24. 1 9 + 5 18
25. 3 4 7 10 26. 7 15 4 9
27. The approximate land area of Jamaica is 4 × 10 3 square miles. The approximate land area of Brazil is 3 × 10 6 square miles. The land area of Brazil is about how many times as large as the land area of Jamaica?
For problems 28 and 29, find the sum or difference.
28. 4.073 + 8.607 + 2.46 29. 65.761 − 34.92
Student Edition: Grade 8, Module 1, Topic B, Lesson 9
Name Date
Writing Equivalent Expressions
Making Unlike Bases Alike
1. Write each expression as a single power with a base of 2.
2. Write each expression as a single power with a base of 5.
For problems 3–6, use the properties and definitions of exponents to write each expression as a single power.
Student Edition: Grade 8, Module 1, Topic B, Lesson 9
Name
Date
1. Use the properties and definitions of exponents to write 6 5 __ 36 as a single power.
2. Simplify (x 0 y −2 z 4 ) 3 . Assume all variables are nonzero.
Student Edition: Grade 8, Module 1, Topic B, Lesson 9
Name Date
Writing Equivalent Expressions
In this lesson, we
• wrote numerical expressions with unlike bases as a single power.
• applied multiple properties and definitions of exponents to write equivalent expressions.
Examples
For problems 1 and 2, apply the properties and definitions of exponents to write the expression as a single power.
Terminology
To simplify an exponential expression, apply as many properties and definitions of exponents as needed to write an equivalent expression containing only positive exponents and the fewest number of bases.
Write
For problems 3–7, simplify. Assume all variables are nonzero.
Simplify the coefficients by writing them as a separate factor.
Use the commutative property to rearrange factors.
5. 8 hk 0 (2h 4 k −3 ) 3
8 hk 0 ( 2 h 4 k 3 ) 3 = 8 ⋅ h ⋅
= 2 3 ⋅ h
⋅ k 9 = 23 ⋅ 23 ⋅ h ⋅ h12 ⋅ k 9
⋅
= 23+3 ⋅ h1+12 ⋅ k 9 = 2 6 h 13 k 9
6. ( m 2 10 n −1 )3
Another way to simplify the numerical bases is 8 ⋅ 2 3 = 8 ⋅ 8 = 64. ( m2 10 n 1 )3 = m 2 3 103 n 1 ⋅ 3 = m6 103 n 3 = 1 103 ⋅ m6 ⋅ n3 = m 6 n3 10 3
7. p 3 r 0 s p −5 r s 4 p 3 r 0 s p 5 r s 4 = p 3 1 s p 5 ⋅ r ⋅ s 4 = p 3 ⋅ s ⋅ p 5 ⋅ r 1 ⋅ s 4 = p 3 ⋅ p 5 ⋅ r 1 ⋅ s ⋅ s 4
= p 3+5 ⋅ r 1 ⋅ s 1+( 4)
= p 8 ⋅ r 1 ⋅ s 3 = p 8 r s 3
Rewrite a power in the denominator with a negative exponent as a power in the numerator with a positive exponent.
Student Edition: Grade 8, Module 1, Topic B, Lesson 9
Name
For problems 1–6, use the properties and definitions of exponents to write the expression as a single power.
For problems 7–16, simplify. Assume all variables are nonzero.
For problems 17–22, match each expression with an equivalent expression. Each answer can be used more than once.
For problems 23–25, fill in the boxes with values that make the equation true. Assume all variables are nonzero.
23. (x ) 2 = 1 x 6
24. 8 m 3 n
2 m 5 n = 4 n2 m 2
25. (5a 3 b )(3a b) = 15ab
26. Which of the following is equal to 12 g h ? Assume g and h are nonzero. Choose all that apply.
A. (4g 2 h)(3g −1 h −2 )
B. 24 g 3 h 5 2 g 2 h 6
C. ( 2 g 3 h 0 ) 2( 3 gh )
D. 3 g 3 h (2 g h) 2
E. 12 h 1 g 1
Remember
For problems 27–30, multiply.
27. 3(1 3 ) 28. 5(2 5 )
29. 7(3 7 ) 30. 4(3 4 )
For problems 31 and 32, identify a value of x that makes the statement true.
31. 7 0 ⋅ 7 x = 7 3 32. (−5) x = 1
33. Find the value of the expression 4.9 × 3.3 ÷ 10.
Student Edition: Grade 8, Module 1, Topic B, Lesson 10
Name Date
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
1. Find the value of 2 4 ⋅ 15 3
Make a Power of 10
For problems 2 and 3, make a power of 10 to evaluate each expression. Write your answer in standard form.
2. 5 2 ⋅ 6 2
5 −3 ⋅ 14 −3 ⋅ 21 3
Write with Fewest Bases
For problems 4 and 5, simplify the expression by using the fewest number of prime bases. 4. 45 −4 ⋅ 15 8
Compare and Connect
For problems 6–8, simplify by using the fewest number of prime bases.
For problems 9–11, evaluate each expression. Write your answer in standard form.
Student Edition: Grade 8, Module 1, Topic B, Lesson 10
Name
1. Simplify the expression 8 7 ⋅ 2 −6 by using the fewest number of prime bases.
2. Make a power of 10 to evaluate 8 2 ⋅ (5 2 ) 3 . Write your answer in standard form.
Student Edition: Grade 8, Module 1, Topic B, Lesson 10
Name Date RECAP
Evaluating Numerical Expressions by Using Properties of Exponents (Optional)
In this lesson, we
• wrote equivalent numerical expressions by using prime factors of the given bases.
• evaluated numerical expressions mentally by making bases of 10.
• simplified numerical expressions by using prime bases.
Examples
1. Make a power of 10 to evaluate the expression 32 ⋅ 50.
Hint: 32 can be written as 2 5 and 50 can be written as 2 ⋅ 25.
To make a base of 10, the 2’s and the 5’s need to have the same exponents. Use the properties of exponents to write 2 5 2 as
4 2 2 Then write as 2 4 2 2
2. Simplify the expression by using the fewest number of prime bases.
Factor 44 into 4 and 11. Because the factor 4 is still not prime, write it as 2 2
Apply the definition of negative exponents to write 11 −2 as 1 11 2
Student Edition: Grade 8, Module 1, Topic B, Lesson 10
Name Date
For problems 1 and 2, make a power of 10 to evaluate the expression. Write your answer in standard form.
1. 2 2 ⋅ 36 ⋅ 5 4
Hint: 36 can be written as 6 2 . 2. 25 2 ⋅ 28 2
Hint: 25 can be written as 5 2 , and 28 can be written as 4 7.
For problems 3–7, simplify the expression by using the fewest number of prime bases.
8. Noor writes 2 3 ⋅ 4 3 ⋅ 8 3 = 64 9 , and Mr. Adams marks it wrong.
a. Explain Noor’s error(s).
b. Simplify 2 3 4 3 8 3 to show Noor how to correctly simplify the expression.
9. Use the properties of exponents to show the steps to simplify
10. Consider the expression 7 ⋅ 49 ⋅ 2 ⋅ 4 ⋅ 14 ⋅ 196.
a. What are some ways can you rewrite the expression by using exponents?
b. Which way do you prefer to rewrite the expression and why?
Remember
For problems 11–14, multiply.
15. Write an equivalent expression for 1 ____ 3854 with a negative exponent.
For problems 16 and 17, find the area of the figure.
Student Edition: Grade 8, Module 1, Topic C
How Many Words Have Ever Been Spoken?
Some questions are so dizzying that we’ll never know the answers—at least not exactly, anyway.
Take this one: “How many words have ever been spoken on Earth?” It’s impossible to know for sure. Nobody has been counting. You don’t even know how many words you’ve spoken today; how could we know the number of words for all people ever?
Well, we can’t. But we can estimate the number of words.
1011
Estimated number of humans who ever lived:
(i.e., 100 billion)
Estimated number of words spoken in an average lifetime:
108
(i.e., 100 million)
Estimated number of words ever spoken on Earth:
1019
(i.e., 10 quintillion)
These numbers are estimates. But that’s okay. With numbers this large, tiny errors don’t matter much. Even big errors may start to seem small by comparison.
Did we miss a billion people? That’s only 1% of our total, which barely changes the answer!
Student Edition: Grade 8, Module 1, Topic C, Lesson 11
Name Date
Small Positive Numbers in Scientific Notation
1. Complete the table as directed during the lesson. The table shows an approximate measurement of various objects.
Approximate Measurement (meters)
Small Positive Numbers
For problems 2–7, write the number in scientific notation or standard form as indicated in the table. Number in Standard Form
in Scientific Notation
0.0007
× 10 −6 6. 0.000 0062
Ordering Numbers in Scientific Notation
8. Order the given numbers from least to greatest in the table. Use the order of magnitude to explain why each number is greater than the number before it.
Number
Numbers Ordered from Least to Greatest
Explanation
Student Edition: Grade 8, Module 1, Topic C, Lesson 11
Name
1. Write 0.0051 in scientific notation.
2. Write 3 × 10 −5 in standard form.
3. Order the numbers from least to greatest.
Student Edition: Grade 8, Module 1, Topic C, Lesson 11
Name Date
Small Positive Numbers in Scientific Notation
In this lesson, we
• used the definition of negative exponents to write small positive numbers in scientific notation.
• ordered numbers written in scientific notation from least to greatest.
Examples 1. Complete the table.
The 5 will be in the place value represented by 10 −4 , which is equal to 1 10 4 , or 1
10,000
The power of 10, 10 −6 , shows the place value of the first digit only, which is 2 The 09 is written after the 2.
The 3 is in the place value represented by 10 −8 . The first factor is 3.2, which is at least 1 but less than 10.
2. The table shows the approximate weights of animals in pounds. Order the animals from heaviest to lightest.
Animal
The blue whale is the heaviest because its weight has the greatest order of magnitude, which is 5
The orders of magnitude are equal for the weights of the African elephant, the hippopotamus, and the giraffe.
In this case, use the first factor of each weight to determine the correct order.
9.5 > 6 > 2.2
Student Edition: Grade 8, Module 1, Topic C, Lesson 11
For problems 1–8, complete the table. Standard Form
3000
Notation
3 × 10 −3
0.0006
× 10 5
0.00045
0.000 007
0.00601
For problems 9–13, compare the numbers by using the < or > symbol. Explain your answer by using the order of magnitude.
14. Before 2006, Pluto was considered to be one of the planets in our solar system. Many people thought it should be excluded from the list of planets because of its size. The table lists the planets, including Pluto, and their approximate diameters in meters.
× 10 6
Name the planets, including Pluto, in order from the smallest diameter to largest diameter.
15. Ava writes 0.00056 in scientific notation as 5.6 × 10 −4 and shows the following work. Explain what Ava is doing in her work.
0. 00056
16. Nora is comparing the diameter of a grain of salt with the length of a flea.
• The average diameter of a grain of salt is 3 × 10 −4 meters.
• The average length of a flea is 1.5 × 10 −3 meters.
Nora believes that the diameter of a grain of salt is greater than the length of a flea because 3 is greater than 1.5. Explain why you agree or disagree with Nora.
17. The mass of a neutron is about 1.67493 × 10 −27 kg. The mass of a proton is about 1.67262 × 10 −27 kg. Explain which is heavier.
Remember
For problems 18–21, multiply.
18(1 3 )
8(1 2 )
45(1 5 )
80( 1 10 )
22. The table represents box office sales of the highest-grossing movies in the years 2017 to 2019. Write each number in scientific notation.
23. The area of the base of a right rectangular prism is 16 square inches. The height of the prism is 4 inches. What is the volume of the prism?
Student Edition: Grade 8, Module 1, Topic C, Lesson 12
Name
Date
Operations with Numbers in Scientific Notation
1. Liam enters 200,000 × 450,000 into his calculator, and the screen shows the following display.
9e+10
Calculate 200,000 × 450,000 by writing the numbers in scientific notation.
Power to a Power
For problems 2–8, use the properties of exponents to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator.
2. (3 × 105)2
3. (5 × 103)2
(2 × 10 −9 ) 2
(2.6 × 10 −4 ) 3
How Many Times
For problems 9–11, use the values from the table to answer the questions. Write the answer in scientific notation. Use a calculator to check your answers.
Object
Grain of Salt (diameter)
Tube of Lip Balm (length)
Whale (length)
(diameter)
(diameter)
Approximate Measurement (meters)
× 10 −4
× 10 −2
× 10
× 10 8
× 10 9
9. A blue whale is about how many times as long as a tube of lip balm?
10. About how many grains of salt positioned side by side would it take to equal the length of a tube of lip balm?
11. Which object from the table is about 50 million times as large as a blue whale?
Student Edition: Grade 8, Module 1, Topic C, Lesson 12
Name Date
1. After a series of calculations, a calculator screen displays this result.
4.399e14
What is the displayed value written in scientific notation?
2. Which number is greater, 93 million or (239,900) 2 ? Show calculations by using scientific notation, and explain your answer.
Student Edition: Grade 8, Module 1, Topic C, Lesson 12
Name Date
Operations with Numbers in Scientific Notation
In this lesson, we
• interpreted scientific notation displayed on digital devices.
RECAP
• used the properties of exponents to efficiently operate with numbers written in scientific notation.
Examples
1. A calculator displays 3.45e−4. Interpret the meaning of this number by writing it in scientific notation and in standard form.
3.45 × 10 −4 = 0.000 345
2. Use the table of approximate animal weights to answer the question.
Animal
Emperor Scorpion
Approximate Weight (pounds)
× 10 −2
Gray Tree Frog 1.6 × 10 −3
Termite 3.3 × 10 −6
An emperor scorpion is about how many times as heavy as a termite? 6.6 × 10 −2 = ⋅ (3.3 × 10 −6 ) 6.6 × 10 2 3 3 × 10
The scorpion weighs more than the termite because the order of magnitude of −2 is greater than the order of magnitude of −6
= 2 × 10 4
An emperor scorpion is about 2 × 10 4 times as heavy as a termite.
3. Is the value of (2.6 × 10 −5 ) 2 greater than or less than 2.6 × 10 −5 ? Explain.
( 2 6 × 10 5) 2 = 2 6 2 × ( 10 5) 2 = 6 76 × 10 10
Use the property of exponents to square both factors of the product within the parentheses.
The value of (2.6 × 10 −5 ) 2 is 6.76 × 10 −10 , which is less than 2.6 × 10 −5 because the order of magnitude of −10 is less than the order of magnitude of −5.
Student Edition: Grade 8, Module 1, Topic C, Lesson 12
For problems 1–4, write the given answer for each calculator display in scientific notation and in standard form.
1. . 678e+ 51 7 2. 5 0.0000023*0.0000 1.15E-10 3. 1.386471e9 4. 1.9321 * 10 18 (1.39 * 10 9 )2
For problems 5–7, use the table of animal weights to answer the questions. Use a calculator to check your answers.
5. A wood mouse is about how many times as heavy as a monarch butterfly?
6. Which animal or insect is about 8 times as heavy as a monarch butterfly?
7. A zebra is about how many times as heavy as a hummingbird?
8. Which number is greater, (1.1 × 10 −3 ) 2 or 1.1 × 10 −3 ? Explain.
9. Which number is less, (3 × 10 6 ) 2 or (6 × 10 3 ) 3 ? Explain.
10. Use the following table from lesson 11 to answer the questions.
a. Comparing only diameters, Jupiter is about how many times as large as Pluto? Round your answer to the nearest whole number.
b. Comparing only diameters, Jupiter is about how many times as large as Mercury? Round your answer to the nearest whole number.
c. Assume you are a voting member of the International Astronomical Union (IAU) and the classification of Pluto is based entirely on the length of its diameter. Would you vote to keep Pluto a planet or to reclassify it as a dwarf planet? Why?
Remember
For problems 11–14, multiply.
15. Simplify the expression 14 11 14 6 . Write the expression with positive exponents.
16. Plot all the points from the table in the coordinate plane.
Student Edition: Grade 8, Module 1, Topic C, Lesson 13
Name Date
Applications with Numbers in Scientific Notation
Operating with Numbers in Scientific Notation
1. On average, an eighth grader’s lungs can hold 3.6 × 10−3 cubic meters of air per breath.
a. What volume of air is in 2 breaths? Write your answer in scientific notation.
b. What volume of air is in 10 breaths? Write your answer in scientific notation.
2. The volume of each breath is about 3.6 × 10−3 cubic meters. How many breaths does it take to reach the goal of filling the balloon with 3.6 × 100 cubic meters of air?
3. Complete the comparison statement.
The volume of the goal is cubic meters. That is times as much as the volume of a single breath, which is cubic meters.
4. On average, an eighth grader breathes 14 times per minute. Assume you have been breathing at about the same rate your whole life.
a. About how many breaths have you taken so far in your life? Write your answer in scientific notation.
b. If the volume of one breath is about 3.6 × 10−3 cubic meters, what is the approximate total volume of breaths you have taken so far in your life? Write your answer in scientific notation.
5. One dollar bill is 0.11 millimeters thick. About how many dollar bills will it take to make a stack 1 meter tall? Write your answer in scientific notation.
6. The moon is approximately 3.84 × 108 meters away from Earth. One dollar bill is 0.11 millimeters thick. About how many dollar bills will it take to make a stack from Earth to the moon? Write your answer in scientific notation.
7. In February 2019, the US national debt was about 22 trillion dollars, or 2.2 × 1013 dollars. Make a comparison statement between the value of the Earth-to-moon stack of dollar bills in problem 6 and the national debt.
Student Edition: Grade 8, Module 1, Topic C, Lesson 13
Name Date EXIT TICKET
The Tokyo Skytree Tower is 6.34 × 10 2 meters tall. A paperclip is about 3.17 × 10 2 meters long. If placed end to end, approximately how many paperclips will it take to measure the Tokyo Skytree Tower?
Student Edition: Grade 8, Module 1, Topic C, Lesson 13
Name Date
Applications with Numbers in Scientific Notation
In this lesson, we
• performed operations with numbers written in scientific notation.
• recognized there are various paths to solve a problem by comparing solution strategies.
Example
Mount Denali, located in Alaska, is the highest mountain peak in North America. Complete parts (a)–(c) to estimate, calculate, and compare the approximate number of pink erasers you must stack to reach the height of Mount Denali.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions.
In the millions
b. The thickness of a pink eraser is approximately 3.1 × 10 −2 feet. The height of Mount Denali is approximately 2.031 × 10 4 feet. Calculate the approximate number of pink erasers that must be stacked to reach the height of Mount Denali.
The definition of negative exponents tells us that
About 6.55 × 10 5 , or 655,000, pink erasers must be stacked to reach the height of Mount Denali.
Student Edition: Grade 8, Module 1, Topic C, Lesson 13
Name Date
PRACTICE
1. Use the following information to fill in the blank. On Earth, there are about 1 quadrillion (1,000,000,000,000,000) ants and about 8 billion people.
On Earth, there are about times as many ants as people.
2. Complete parts (a) and (b) to estimate and calculate the approximate number of sheets of paper you must stack to reach the height of Mount Everest.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions.
b. A stack of 200 sheets of paper measures approximately 5 × 10 −2 feet. The height of Mount Everest is approximately 2.9035 × 10 4 feet. Calculate the approximate number of sheets of paper you must stack to reach the height of Mount Everest.
3. Complete parts (a) and (b) to estimate and calculate the approximate number of people it would take standing shoulder to shoulder to fill the entire United States.
a. Estimate by making an educated guess about the place value of the answer. For example, describe the place value as in the thousands or in the tens of millions.
b. The United States has a land area of about 3.797 × 10 6 square miles. Each person takes up about 4 square feet of area. Calculate the number of people that could stand shoulder to shoulder to fill the United States. (1 square mile ≈ 2.788 × 10 7 square feet)
Remember
For problems 4–7, multiply.
4. 4(5 8 ) 5. 4( 7 24 )
6. 11( 7 33 ) 7. 10(14 80 )
8. Simplify (x −6 y 5 z 0 ) −4 . Assume all variables are nonzero.
9. The length of a line segment is 12 units. One endpoint of the line segment is (−3, 6). Find four points that could be the other endpoint of the line segment. Use the coordinate plane as needed.
Student Edition: Grade 8, Module 1, Topic C, Lesson 14
Name Date
Choosing Units of Measurement
For problems 1–3, choose a unit of measurement to use for each problem. Then explain why you chose that unit of measurement.
1. You wonder how long you can listen to your favorite music before you hear the same song twice. You have 1000 songs, and the average song length is 4 minutes.
Would you want to display the time in minutes, hours, or days of music?
2. You wonder about the weight of all the oranges used for commercial production in the United States in a year. Your research indicates that each box of oranges weighs about 85 pounds, and about 92 million boxes of oranges were used for commercial production in 2017 and 2018.
Would you want to display the weight in ounces, pounds, or tons of oranges?
3. The seafloor spreads at a rate of approximately 10 centimeters per year. You collect data on the spread of the seafloor each week.
Would you want to record the data in millimeters, centimeters, or meters?
Seconds of Life
4. Mr. Jacobs just welcomed a baby into his family. Since birth, the baby has been alive for 1 million seconds. Mr. Jacobs has been alive for 1 billion seconds. How many seconds older is Mr. Jacobs than the baby? Write the answer in scientific notation.
Two-Way Radio Problem
5. Consider the given information about Henry’s Handhelds and Winnie’s Walkie Talkies. Create a persuasive advertisement for either company by using a comparison statement. Use the space provided to show your work in making the comparison statement.
Use this space to write your advertisement comparison statement.
