For nonzero real numbers [latex]a[/latex] and [latex]b[/latex] and integers [latex]m[/latex] and [latex]n[/latex]
scientific notation a shorthand notation for writing very large or very small numbers in the form [latex]a\times {10}^{n}[/latex] where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer
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Scientific notation is a smart way of writing huge whole numbers and too small decimal numbers. This page contains worksheets based on rewriting whole numbers or decimals in scientific notation and rewriting scientific notation form to standard form. This set of printable worksheets is specially designed for students of grade 6, grade 7, grade 8, and high school. Access some of them for free!
Express in Scientific Notation
Each pdf worksheet contains 14 problems rewriting whole numbers to scientific notation. Easy level has whole numbers up to 5-digits; Moderate level has more than 5-digit numbers.
Express in Standard Notation
Students of 6th grade need to express each scientific notation in standard notation. An example is provided in each worksheet.
Both Standard and Scientific Notations
Each printable worksheet contains expressing numbers in both scientific and standard form.
Rewrite in Scientific Notation
Rewrite the given decimals in scientific notation. Move the decimal point to the left until you get the first non-zero digit. The number of steps you moved represent the power (index) of 10.
Rewrite in Standard Notation
In this set of pdf worksheets, express each number in standard notation. Easy level has indices more than -5; Moderate level has indices less than -4.
Each worksheet has ten problems expressing decimals in both standard and scientific notation.
Convert to Scientific Notation
The printable worksheets in this section contain expressing both whole numbers and decimals in scientific notation.
Convert to Standard Notation
The exponent in each scientific notation can be either positive or negative.
This section of pdf worksheets gives the complete review in rewriting numbers in both standard and scientific notation. Both positive and negative exponents included.
Comparing Numbers in Scientific Notation
Call upon your inner math maestro as you sail through figuring out which of the given numbers in scientific notation is greater than, less than, or equal to the other.
Addition and Subtraction
This section reinforces the knowledge in adding and subtracting numbers in scientific notation.
Multiplication and Division
Use laws of exponents (indices) to multiply and divide the expressions. Express the final answer in scientific notation.
Simplify the Expression
Each worksheet gives the complete review in performing operations with scientific notations, making it ideal for 7th grade, 8th grade, and high school students.
Related Worksheets
» Exponents
» Logarithms
» Multiplying Decimals by Powers of Ten
» Dividing Decimals by Powers of Ten
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Maneuvering the Middle
Student-Centered Math Lessons
Let’s chat about scientific notation and exponents ! I have found that the simplest skills in math are often the most miscalculated and confusing for students. Exponents and scientific notation can fall into this trap.
It is important with both exponents and scientific notation that students understand that they show a different way to represent a value.
Before even showing an exponent, start by showing the expanded form like 7*7*7*7*7. You can start by asking students:
Tip: I have learned the hard way to NEVER use 2^2 in any of your early examples because it will just confuse students into thinking you multiply the base and exponent.
Easy, no-plan activity idea: A fun way for students to practice exponents is by using concentric circles. The inside circle is the base and the outside circle is the exponent. Assign students numbers 1-10 and have them rotate to a new partner each round. Students pair up, write down the exponent form, the expanded form, and then calculate the standard form. Keep rotating until your time is up.
The laws of exponents are so fun! I love how students can build on their previous knowledge to come up with the laws themselves. For example:
On a Facebook thread, I recently saw a teacher say, “When in doubt, expand it out.” If a student forgets a law, all they need to do is expand it and calculate to discover the law again. That is something a student is more likely to do if you are modeling it consistently.
Because the laws are so accessible, this content really shines as a discovery-based lesson. Your students could also participate in a jigsaw. Each group becomes experts at their assigned law, then they present the law, the proof, and examples to their peers.
If you go the traditional teaching route, I recommend splitting this skill up over at least 2 days. Maneuvering the Middle 8th grade curriculum covers multiplying/dividing like bases, power to power, and product to power on day one. Negative and zero exponents are covered on day two.
I highly recommend an anchor chart with all of the laws for easy reference. Sometimes in my last class on a Friday, my brain needed to look at an anchor chart to give it the boost it needed (and I am the teacher).
Like I said before, scientific notation is just a different way to represent a value. Here is a great way to introduce why we might use scientific notation. Write down the mass of Earth and Mars on your whiteboard or project it. Make sure students will have to copy it down themselves since that is part of your point.
Start by asking students to read the numbers to you. You will get some funny responses. Then ask students to add them or subtract the masses. As students write and count all of the zeros and inevitably miscalculate or miscount the number of zeros, you can introduce why we used scientific notation. (Less room for error, more efficient) Scientific notation is similar to typing TTYL instead of typing “talk to you later.”
Avoid using right or left when describing the direction to move the decimal. Instead, emphasize that smaller numbers will have negative exponents and larger numbers will have positive exponents. This re-enforces the negative exponent law.
Speaking of exponent laws, scientific notation operations reinforce the laws of exponents. If you look at the vertical alignment, scientific notation only shows up in 8th grade (in TEKS and CCSS), so at least, it reinforces other important concepts that students will use in Algebra 1 and 2.
I have never (and will never) teach Science, but it did occur to me to look up the Texas Science standards, and take a look at this chemistry standard –
“C.2(G) express and manipulate chemical quantities using scientific conventions and mathematical procedures, including dimensional analysis, scientific notation , and significant figures”
An opportunity for cross curricular?! Wahoo! If this is something that has peaked your interest, here is a NASA themed exploration lesson with resources for practicing scientific notation. This demos activity is also a great science based activity.
What tips do you have for teaching exponents and scientific notation?
Digital Activities for 6th - 8th grade Math & Algebra 1 interactive | easy-to-use with Google Slides | self-grading Google Forms exit ticket
Algebra 2 exponent worksheets.
Ultimate math solver (free) free algebra solver ... type anything in there, popular pages @ mathwarehouse.com.
Get Free Access to Download Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation PDF from here. Start your preparation with the help of Go Math Grade 8 Answer Key . It is essential for all the students to learn the concepts of this chapter in-depth. So, make use of the Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Solution Key links and go through the solutions.
Check out the list of the topics before you start your preparation. You can step by step explanation for all the questions in HMH Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation for free of cost. Quickly download Go Math Grade 8 Chapter 2 Answer Key PDF and fix the timetable to prepare.
Lesson 1: Integer Exponents
Lesson 2: Scientific Notation with Positive Powers of 10
Lesson 3: Scientific Notation with Negative Powers of 10
Lesson 4: Operations with Scientific Notation
Mixed Review
Find the value of each power.
Question 1. 8 −1 = \(\frac{□}{□}\)
Answer: \(\frac{1}{8}\)
Explanation: Base = 8 Exponent = 1 8 −1 = (1/8) 1 = 1/8
Question 2. 6 −2 = \(\frac{□}{□}\)
Answer: \(\frac{1}{36}\)
Explanation: Base = 6 Exponent = 2 6 −2 = (1/6) 2 = 1/36
Question 3. 256 0 = ______
Explanation: 256 0 Base = 256 Exponent = 0 Anything raised to the zeroth power is 1. 256 0 = 1
Question 4. 10 2 = ______
Answer: 100
Explanation: Base = 10 Exponent = 2 10 2 = 10 × 10 = 100
Question 5. 5 4 = ______
Answer: 625
Explanation: Base = 5 Exponent = 4 5 4 = 5 × 5 × 5 × 5 = 625
Question 6. 2 −5 = \(\frac{□}{□}\)
Answer: \(\frac{1}{32}\)
Explanation: Base = 2 Exponent = 5 2 −5 = (1/2) 5 = (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/32
Question 7. 4 −5 = \(\frac{□}{□}\)
Answer: \(\frac{1}{1,024}\)
Explanation: Base = 4 Exponent = 5 4 −5 = (1/4) 5 = (1/4) × (1/4) × (1/4) × (1/4) × (1/4) = 1/1,024
Question 8. 89 0 = ______
Explanation: 89 0 Base = 89 Exponent = 0 Anything raised to the zeroth power is 1. 89 0 = 1
Question 9. 11 −3 = \(\frac{□}{□}\)
Answer: \(\frac{1}{1,331}\)
Explanation: Base = 11 Exponent = 3 11 −3 = (1/11) 3 = (1/11) × (1/11) × (1/11) = 1/1,331
Use properties of exponents to write an equivalent expression.
Question 10. 4 ⋅ 4 ⋅ 4 = 4 ? Type below: _____________
Answer: 4 3
Explanation: The same number 4 is multiplying 3 times. The number of times a term is multiplied called the exponent. So the base is 4 and the exponent is 3 4 ⋅ 4 ⋅ 4 = 4 3
Question 11. (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2 ? ⋅ 2 ? = 2 ? Type below: _____________
Answer: 2 5
Explanation: The same number 2 is multiplying 5 times. The number of times a term is multiplied called the exponent. So the base is 2 and the exponent is 5 (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2 2 ⋅ 2 3 = 2 5
Question 12. \(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\) = 6 ? Type below: _____________
Answer: 6 2
Explanation: \(\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } } \) = \(\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}\) Cancel the common factors 6.6 Base = 6 Exponent = 2 6 2
Question 13. \(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \) = 8 ?-? = 8 ? Type below: _____________
Answer: 8 3
Explanation: \(\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } } \) Bases are common. So, the exponents are subtracted 8 12-9 = 8 3
Question 14. 5 10 ⋅ 5 ⋅ 5 = 5 ? Type below: _____________
Answer: 5 12
Explanation: Bases are common and multiplied. So, the exponents are added Base = 5 Exponents = 10 + 1 + 1 = 12 5 12
Question 15. 7 8 ⋅ 7 5 = 7 ? Type below: _____________
Answer: 7 13
Explanation: Bases are common and multiplied. So, the exponents are added Base = 7 Exponents = 8 + 5 = 13 7 13
Question 16. (6 2 ) 4 = (6 ⋅ 6) ? = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (? ⋅ ?) ⋅ ? = 6 ? Type below: _____________
Answer: 6 8
Explanation: (6 2 ) 4 = (6 ⋅ 6) 4 = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) = 6 2 ⋅ 6 2 . 6 2 ⋅ 6 2 Bases are common and multiplied. So, the exponents are added = 6 2+2+2+2 6 8
Question 17. (3 3 ) 3 = (3 ⋅ 3 ⋅ 3) 3 = (3 ⋅ 3 ⋅ 3) ⋅ (? ⋅ ? ⋅ ?) ⋅ ? = 3 ? Type below: ______________
Answer: 3 9
Explanation: (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) = 3 3 ⋅ 3 3 ⋅ 3 3 Bases are common and multiplied. So, the exponents are added 3 3 + 3 + 3 3 9
Simplify each expression.
Question 18. (10 − 6) 3 ⋅4 2 + (10 + 2) 2 ______
Answer: 1,168
Explanation: 4³. 4² + (12)² = 4 5 + (12)² = 4 5 + (12 . 12)² 4 5 + (144) = 1,024 + 144 = 1,168
Question 19. \(\frac { { (12-5) }^{ 7 } }{ { [(3+4)^{ 2 }] }^{ 2 } } \) ________
Answer: 343
Explanation: 7 7 ÷ (7²)² = 7 7 ÷ 7 4 7 7-4 7³ 7 . 7 . 7 = 343
ESSENTIAL QUESTION CHECK-IN
Question 20. Summarize the rules for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power. Type below: ______________
Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. The “power rule” tells us that to raise a power to a power, just multiply the exponents.
Question 21. Explain why the exponents cannot be added in the product 12 3 ⋅ 11 3 . Type below: ______________
Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The bases are not the same in the given problem. => (12)³ x (11)³ If we solve this equation following the rule of exponent will get the correct answer: => (12 x 12 x 12) x (11 x 11 x 11) => 1728 X 1331 => the answer is 2 299 968 But if we add the exponent, the answer would be wrong => (12)³ x (11)³ => 132^6 => 5289852801024 which is wrong.
Question 22. List three ways to express 3 5 as a product of powers. Type below: ______________
Answer: 3¹ . 3 4 3² . 3 3 3³ . 3 2
Question 23. Astronomy The distance from Earth to the moon is about 22 4 miles. The distance from Earth to Neptune is about 22 7 miles. Which distance is the greater distance and about how many times greater is it? _______ times
Answer: (22)³ or 10,648 times
Explanation: The distance from Earth to the moon is about 22 4 miles. The distance from Earth to Neptune is about 22 7 miles. 22 7 – 22 4 = (22)³ The greatest distance is from Earth to Neptune The distance from Earth to Neptune is greater by (22)³ or 10,648 miles
Question 24. Critique Reasoning A student claims that 8 3 ⋅ 8 -5 is greater than 1. Explain whether the student is correct or not. ______________
Answer: 8 3 ⋅ 8 -5 is = 8 -2 (1/8)² (1/8) . (1/8) = 1/64 = 0.015 The student is not correct.
Find the missing exponent.
Question 25. (b 2 ) ? = b -6 _______
Answer: (b 2 ) -8
Explanation: (b 2 ) ? = b -6 (b -6 ) = b 2-8 (b 2-8 ) = b 2 . b -8 (b 2 ) -8 = b -6
Question 26. x ? ⋅ x 6 = x 9 _______
Explanation: x ? ⋅ x 6 = x 9 x 9 = x 3 + 6 x³ x 6
Question 27. \(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y 6 _______
Answer: y25 ÷ y16
Explanation: \(\frac { { y }^{ 25 } }{ { y }^{ ? } } \) = y 6 y 6 = y 25 – 16 y 25 ÷ y 16
Question 28. Communicate Mathematical Ideas Why do you subtract exponents when dividing powers with the same base? Type below: ______________
Answer: To divide exponents (or powers) with the same base, subtract the exponents. The division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.
Question 29. Astronomy The mass of the Sun is about 2 × 10 27 metric tons, or 2 × 10 30 kilograms. How many kilograms are in one metric ton? ________ kgs in one metric ton
Answer: 1,000 kgs in one metric ton
Explanation: The mass of the Sun is about 2 × 10 27 metric tons, or 2 × 10 30 kilograms. 2 × 10 27 metric tons = 2 × 10 30 ki 1 metric ton = 2 × 10 30 ki ÷ 2 × 10 27 = (10)³ = 1,000 kgs in one metric ton
Question 30. Represent Real-World Problems In computer technology, a kilobyte is 2 10 bytes in size. A gigabyte is 2 30 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte? Type below: ______________
Answer: 2 40 bytes
Explanation: In computer technology, a kilobyte is 2 10 bytes in size. A gigabyte is 2 30 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. terabyte = 2 10 bytes × 2 30 bytes = 2 10+30 bytes = 2 40 bytes
Question 31. Write equivalent expressions for x 7 ⋅ x -2 and \(\frac { { x }^{ 7 } }{ { x }^{ 2 } } \). What do you notice? Explain how your results relate to the properties of integer exponents. Type below: ______________
Answer: x^a * x^b = x^(a+b) and x^-a = 1/x^a Therefore, x^7 * x^-2 = x^7/x^2 = x^5 or x^7 * x^-2 = x^(7-2) = x^5 x^7 / x^2 = x^7 * x^-2
Question 32. Look for a Pattern Describe any pattern you see in the table. Type below: ______________
Answer: As the number of rows increased, the number of cubes in each row by multiple of 3.
Question 33. Using exponents, how many cubes will be in Row 6? How many times as many cubes will be in Row 6 than in Row 3? _______ times more cubes
Answer: (3 3 ) times more cubes
Explanation: For row 6, the number of cubes in each row = (3 6 ) (3 6 ) ÷ (3 3 ) = (3 6-3 ) = (3 3 ) (3 3 ) times more cubes
Question 34. Justify Reasoning If there are 6 rows in the triangle, what is the total number of cubes in the triangle? Explain how you found your answer. ______ cubes
Answer: 1,092 cubes
Explanation: (3 1 ) + (3 2 ) + (3 3 ) + (3 4 ) + (3 5 ) + (3 6 ) 3 + 9 + 27 + 81 + 243 + 729 = 1,092
Focus on Higher Order Thinking
Question 35. Critique Reasoning A student simplified the expression \(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \) as \(\frac{1}{3}\). Do you agree with this student? Explain why or why not. ______________
Answer: \(\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } } \) (6 2 ) ÷ (6 2 )² (6 2 ) ÷ (6 4 ) (6 2 – 4 ) (6 -2 ) = 1/36 I don’t agree with the student
Question 36. Draw Conclusions Evaluate –a n when a = 3 and n = 2, 3, 4, and 5. Now evaluate (–a) n when a = 3 and n = 2, 3, 4, and 5. Based on this sample, does it appear that –a n = (–a) n ? If not, state the relationships, if any, between –a n and (–a) n . Type below: ______________
Answer: –a n when a = 3 and n = 2, 3, 4, and 5. -3 n -(3 2 )= -9 (–a) n = -3 . -3 = 9 –a n = (–a) n are not equal.
Question 37. Persevere in Problem Solving A number to the 12th power divided by the same number to the 9th power equals 125. What is the number? _______
Answer: Let’s call our number a. (a 12 ) ÷ (a 9 ) (a 12-9 ) = (a 3 ) (a 3 ) = 125 a = (125) 1/3 a = 5
Write each number in scientific notation.
Question 1. 58,927 (Hint: Move the decimal left 4 places) Type below: ______________
Answer: 5.8927 × (10) 4
Explanation: 58,927 Move the decimal left 4 places 5.8927 × (10) 4
Question 2. 1,304,000,000 (Hint: Move the decimal left 9 places.) Type below: ______________
Answer: 1.304 × (10) 9
Explanation: 1,304,000,000 Move the decimal left 9 places 1.304 × (10) 9
Question 3. 6,730,000 Type below: ______________
Explanation: 6,730,000 Move the decimal left 6 places 6.73 × (10) 6
Question 4. 13,300 Type below: ______________
Explanation: 13,300 Move the decimal left 4 places 1.33 × (10) 4
Question 5. An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms. Type below: ______________
Explanation: 97,700,000,000,000,000,000,000 Move the decimal left 22 places 9.77 × (10) 22
Question 6. The distance from Earth to the Moon is about 384,000 kilometers. Type below: ______________
Answer: 3.84 × (10) 6
Explanation: 384,000 Move the decimal left 6 places 3.84 × (10) 6
Write each number in standard notation.
Question 7. 4 × 10 5 (Hint: Move the decimal right 5 places.) Type below: ______________
Answer: 400,000
Explanation: 4 × 10 5 Move the decimal right 5 places 400,000
Question 8. 1.8499 × 10 9 (Hint: Move the decimal right 9 places.) Type below: ______________
Answer: 1849900000
Explanation: 1.8499 × 10 9 Move the decimal right 9 places 1849900000
Question 9. 6.41 × 10 3 Type below: ______________
Answer: 6410
Explanation: 6.41 × 10 3 Move the decimal right 3 places 6410
Question 10. 8.456 × 10 7 Type below: ______________
Answer: 84560000
Explanation: 8.456 × 10 7 Move the decimal right 7 places 84560000
Question 11. 8 × 10 5 Type below: ______________
Answer: 800,000
Explanation: 8 × 10 5 Move the decimal right 5 places 800,000
Question 12. 9 × 10 10 Type below: ______________
Answer: 90000000000
Explanation: 9 × 10 10 Move the decimal right 10 places 90000000000
Question 13. Diana calculated that she spent about 5.4 × 10 4 seconds doing her math homework during October. Write this time in standard notation. Type below: ______________
Answer: 5400
Explanation: Diana calculated that she spent about 5.4 × 10 4 seconds doing her math homework during October. 5.4 × 10 4 Move the decimal right 4 places 5400
Question 14. The town recycled 7.6 × 10 6 cans this year. Write the number of cans in standard notation Type below: ______________
Answer: 7600000
Explanation: The town recycled 7.6 × 10 6 cans this year. 7.6 × 10 6 Move the decimal right 10 places 7600000
Question 15. Describe how to write 3,482,000,000 in scientific notation. Type below: ______________
Answer: 3.482 × (10) 9
Explanation: 3,482,000,000 Move the decimal left 9 places 3.482 × (10) 9
Paleontology
Question 16. Apatosaurus ______________ Type below: ______________
Answer: 6.6 × (10) 4
Explanation: 66,000 Move the decimal left 4 places 6.6 × (10) 4
Question 17. Argentinosaurus ___________ Type below: ______________
Answer: 2.2 × (10) 5
Explanation: 220,000 Move the decimal left 5 places 2.2 × (10) 5
Question 18. Brachiosaurus ______________ Type below: ______________
Answer: 1 × (10) 5
Explanation: 100,000 Move the decimal left 5 places 1 × (10) 5
Question 19. Camarasaurus ______________ Type below: ______________
Answer: 4 × (10) 4
Explanation: 40,000 Move the decimal left 4 places 4 × (10) 4
Question 20. Cetiosauriscus ____________ Type below: ______________
Answer: 1.985 × (10) 4
Explanation: 19,850 Move the decimal left 4 places 1.985 × (10) 4
Question 21. Diplodocus _____________ Type below: ______________
Answer: 5 × (10) 4
Explanation: 50,000 Move the decimal left 4 places 5 × (10) 4
Question 22. A single little brown bat can eat up to 1,000 mosquitoes in a single hour. Express in scientific notation how many mosquitoes a little brown bat might eat in 10.5 hours. Type below: ______________
Answer: 1.05 × (10) 4
Explanation: (1000 x 10.5) = 10500. The little brown bat can eat 10500 mosquitoes in 10.5 hours. 1.05 × (10) 4
Question 23. Multistep Samuel can type nearly 40 words per minute. Use this information to find the number of hours it would take him to type 2.6 × 10 5 words. Type below: ______________
Answer: Samuel can type 40 words per minute. Then how many hours will it take for him to type 2.6 words times 10 to the power of five words 2.6 words time 10 to the power of 5 2.6 × (10) 4 2.6 x 100 000 = 260 000 words in all. Now, we need to find the number of words Samuel can type in an hour 40 words/minutes, in 1 hour there are 60 minutes 40 x 60 2,400 words /hour Now, let’s divide the total of words he needs to type to the number of words he can type in an hour 260 000 / 2 400 108.33 hours.
Question 24. Entomology A tropical species of mite named Archegozetes longisetosus is the record holder for the strongest insect in the world. It can lift up to 1.182 × 10 3 times its own weight. a. If you were as strong as this insect, explain how you could find how many pounds you could lift. Type below: ______________
Answer: Number of pounds you can lift by 1.182 × 10 3 by your weight
Question 24. b. Complete the calculation to find how much you could lift, in pounds, if you were as strong as an Archegozetes longisetosus mite. Express your answer in both scientific notation and standard notation. Type below: ______________
Answer: scientific notation: 1.182 × 10 5 standard notation: 118200
Explanation: 1.182 × 10 3 × 10 2 1.182 × 10 5 118200
Question 25. During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. In scientific notation, express approximately how much the calves weighed all together. Type below: ______________
Answer: 4.6 × 10 3
Explanation: During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. Total weight of the claves = 230 × 20 = 4600 Move the decimal left 3 places 4.6 × 10 3
Question 26. Classifying Numbers Which of the following numbers are written in scientific notation? 0.641 × 10 3 9.999 × 10 4 2 × 10 1 4.38 × 5 10 Type below: ______________
Answer: 0.641 × 10 3 4.38 × 5 10
Question 27. Explain the Error Polly’s parents’ car weighs about 3500 pounds. Samantha, Esther, and Polly each wrote the weight of the car in scientific notation. Polly wrote 35.0 × 10 2 , Samantha wrote 0.35 × 10 4 , and Esther wrote 3.5 × 10 4 . a. Which of these girls, if any, is correct? ______________
Answer: None of the girls is correct
Question 27. b. Explain the mistakes of those who got the question wrong. Type below: ______________
Answer: Polly did not express the number such first part is greater than or equal to 1 and less than 10 Samantha did not express the number such first part is greater than or equal to 1 and less than 10 Esther did not express the exponent of 10 correctly
Question 28. Justify Reasoning If you were a biologist counting very large numbers of cells as part of your research, give several reasons why you might prefer to record your cell counts in scientific notation instead of standard notation. Type below: ______________
Answer: It is easier to comprehend the magnitude of large numbers when in scientific notation as multiple zeros in the number are removed and express as an exponent of 10. It is easier to compare large numbers when in scientific notation as numbers are be expressed as a product of a number greater than or equal to 1 and less than 10 It is easier to multiply the numbers in scientific notation.
Question 29. Draw Conclusions Which measurement would be least likely to be written in scientific notation: number of stars in a galaxy, number of grains of sand on a beach, speed of a car, or population of a country? Explain your reasoning. Type below: ______________
Answer: speed of a car
Explanation: As we know scientific notation is used to express measurements that are extremely large or extremely small. The first two are extremely large, then, they could be expressed in scientific notation. If we compare the speed of a car and the population of a country, it is clear that the larger will be the population of a country. Therefore, it is more likely to express that in scientific notation, so the answer is the speed of a car.
Question 30. Analyze Relationships Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first. 4.5 × 10 6 2.1 × 10 8 Type below: ______________
Answer: 2.1 × 10 8
Explanation: 2.1 × 10 8 is greater because the power of 10 is greater in 2.1 × 10 8
Question 31. Communicate Mathematical Ideas To determine whether a number is written in scientific notation, what test can you apply to the first factor, and what test can you apply to the second factor? Type below: ______________
Answer: The first term must have one number before the decimal point the second term (factor) must be 10 having some power.
Question 1. 0.000487 Hint: Move the decimal right 4 places. Type below: ______________
Answer: 4.87 × 10 -4
Explanation: 0.000487 Move the decimal right 4 places 4.87 × 10 -4
Question 2. 0.000028 Hint: Move the decimal right 5 places Type below: ______________
Answer: 2.8 × 10 -5
Explanation: 0.000028 Move the decimal right 5 places 2.8 × 10 -5
Question 3. 0.000059 Type below: ______________
Answer: 5.9 × 10 -5
Explanation: 0.000059 Move the decimal right 5 places 5.9 × 10 -5
Question 4. 0.0417 Type below: ______________
Answer: 4.17 × 10 -2
Explanation: 0.0417 Move the decimal right 2 places 4.17 × 10 -2
Question 5. Picoplankton can be as small as 0.00002 centimeters. Type below: ______________
Answer: 2 × 10 -5
Explanation: 0.00002 Move the decimal right 5 places 2 × 10 -5
Question 6. The average mass of a grain of sand on a beach is about 0.000015 gram. Type below: ______________
Answer: 1.5 × 10 -5
Explanation: 0.000015 Move the decimal right 5 places 1.5 × 10 -5
Question 7. 2 × 10 -5 Hint: Move the decimal left 5 places. Type below: ______________
Answer: 0.00002
Explanation: 2 × 10 -5 Move the decimal left 5 places 0.00002
Question 8. 3.582 × 10 -6 Hint: Move the decimal left 6 places. Type below: ______________
Answer: 0.000003582
Explanation: 3.582 × 10 -6 Move the decimal left 6 places 0.000003582
Question 9. 8.3 × 10 -4 Type below: ______________
Answer: 0.00083
Explanation: 8.3 × 10 -4 Move the decimal left 4 places 0.00083
Question 10. 2.97 × 10 -2 Type below: ______________
Answer: 0.0297
Explanation: 2.97 × 10 -2 Move the decimal left 2 places 0.0297
Question 11. 9.06 × 10 -5 Type below: ______________
Answer: 0.0000906
Explanation: 9.06 × 10 -5 Move the decimal left 5 places 0.0000906
Question 12. 4 × 10 -5 Type below: ______________
Answer: 0.00004
Explanation: 4 × 10 -5 Move the decimal left 5 places 0.00004
Question 13. The average length of a dust mite is approximately 0.0001 meters. Write this number in scientific notation. Type below: ______________
Answer: 1 × 10 -4
Explanation: The average length of a dust mite is approximately 0.0001 meters. 0.0001 Move the decimal right 4 places 1 × 10 -4
Question 14. The mass of a proton is about 1.7 × 10 -24 grams. Write this number in standard notation. Type below: ______________
Answer: 0.000000000000000000000017
Explanation: The mass of a proton is about 1.7 × 10 -24 grams. 1.7 × 10 -24 Move the decimal left 24 places 0.000000000000000000000017
Question 15. Describe how to write 0.0000672 in scientific notation. Type below: ______________
Answer: 6.72 × 10 -5
Explanation: 0.0000672 Move the decimal right 5 places 6.72 × 10 -5
Question 16. Alpaca _______ Type below: ______________
Answer: 2.77 × 10 -3
Explanation: 0.00277 Move the decimal right 3 places 2.77 × 10 -3
Question 17. Angora rabbit _____________ Type below: ______________
Answer: 1.3 × 10 -3
Explanation: 0.0013 Move the decimal right 3 places 1.3 × 10 -3
Question 18. Llama ____________ Type below: ______________
Answer: 3.5 × 10 -3
Explanation: 0.0035 Move the decimal right 3 places 3.5 × 10 -3
Question 19. Angora goat ____________ Type below: ______________
Answer: 4.5 × 10 -3
Explanation: 0.0045 Move the decimal right 3 places 4.5 × 10 -3
Question 20. Orb web spider ___________ Type below: ______________
Answer: 1.5 × 10 -2
Explanation: 0.015 Move the decimal right 2 places 1.5 × 10 -2
Question 21. Vicuña __________ Type below: ______________
Answer: 8 × 10 -4
Explanation: 0.0008 Move the decimal right 4 places 8 × 10 -4
Question 22. Make a Conjecture Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning. Type below: ______________
Answer: The ounces in a cup of milk would be least likely to be written in scientific notation. The ounces in a cup of milk is correct. Scientific notation is used for either very large or extremely small numbers. The thickness of dog hair is very small as the hair is thin. Hence can be converted to scientific notation. The radius of a period on this page is also pretty small. Hence can be converted to scientific notation. The ounces in a cup of milk. There are 8 ounces in a cup, so this is least likely to be written in scientific notation.
Question 23. Multiple Representations Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation Type below: ______________
Answer: 7 centimeters convert to meters In every 1 meter, there are 100 centimeters = 7/100 = 0.07 Therefore, in 7 centimeters there are 0.07 meters. 7 cm is a whole number while 0.07 m is a decimal number Scientific Notation of each number 7 cm = 7 x 10° 7 m = 1 x 10¯² Scientific notation, by the way, is an expression used by the scientist to make a large number of very small number easy to handle.
Question 24. Draw Conclusions A graphing calculator displays 1.89 × 10 12 as 1.89E12. How do you think it would display 1.89 × 10 -12 ? What does the E stand for? Type below: ______________
Answer: 1.89E-12. E= Exponent
Explanation:
Question 25. Communicate Mathematical Ideas When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1? Type below: ______________
Answer: A number written in scientific notation is of the form a × 10 -n where 1 ≤ a < 10 and n is an integer The number is greater than or equal to one if n ≥ 0.
Question 26. The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation. Type below: ______________
Answer: 4.7 × 10 -5
Explanation: The volume of a drop of a certain liquid is 0.000047 liter. Move the decimal right 5 places 4.7 × 10 -5
Question 27. Justify Reasoning If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response. ______________
Answer: Negative
Explanation: Scientific notation is used to express very small or very large numbers. Very small numbers are written in scientific notation using negative exponents. Very large numbers are written in scientific notation using positive exponents. Since a ladybug is very small, we would use the very small scientific notation, which uses negative exponents.
Question 28. Type below: ______________
Answer: 1.74 × (10) 6
Explanation: The moon = 1,740,000 Move the decimal left 6 places 1.74 × (10) 6
Question 29. Type below: ______________
Answer: 1.25e-10
Explanation: 1.25 × (10) -10 Move the decimal left 10 places 1.25e-10
Question 30. Type below: ______________
Answer: 2.8 × (10) 3
Explanation: 0.0028 Move the decimal left 3 places 2.8 × (10) 3
Question 31. Type below: ______________
Answer: 71490000
Explanation: 7.149 × (10) 7 Move the decimal left 7 places 71490000 Question 32. Type below: ______________
Answer: 1.82 × (10) -10
Explanation: 0.000000000182 Move the decimal right 10 places 1.82 × (10) -10
Question 33. Type below: ______________
Answer: 3397000
Explanation: 3.397 × (10) 6 Move the decimal left 6 places 3397000
Question 34. List the items in the table in order from the smallest to the largest. Type below: ______________
Answer: 1.82 × (10) -10 1.25 × (10) -10 2.8 × (10) 3 1.74 × (10) 6 3.397 × (10) 6 7.149 × (10) 7
Question 35. Analyze Relationships Write the following diameters from least to greatest. 1.5 × 10 -2 m ; 1.2 × 10 2 m ; 5.85 × 10 -3 m ; 2.3 × 10 -2 m ; 9.6 × 10 -1 m. Type below: ______________
Answer: 5.85 × 10 -3 m, 1.5 × 10 -2 m, 2.3 × 10 -2 m, 9.6 × 10 -1 m, 1.2 × 10 2 m
Explanation: 1.5 × 10 -2 m = 0.015 1.2 × 10 2 m = 120 5.85 × 10 -3 m = 0.00585 2.3 × 10 -2 m = 0.023 9.6 × 10 -1 m = 0.96 0.00585, 0.015, 0.023, 0.96, 120
Question 36. Critique Reasoning Jerod’s friend Al had the following homework problem: Express 5.6 × 10 -7 in standard form. Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it? Type below: ______________
Explanation: 5.6 × 10 -7 in 0.000000056 Al wrote 56,000,000. AI wrote the zeroes to the right side of the 56 which is not correct. As the exponent of 10 is negative zero’s need to add to the left of the number.
Question 37. Make a Conjecture Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer. Type below: ______________
Answer: When the division is performed, the denominator exponent is subtracted from the numerator exponent. Subtracting a negative value from the numerator exponent will increase its value.
Add or subtract. Write your answer in scientific notation.
Question 1. 4.2 × 10 6 + 2.25 × 10 5 + 2.8 × 10 6 4.2 × 10 6 + ? × 10 ? + 2.8 × 10 6 4.2 + ? + ? ? × 10? Type below: ______________
Answer: 4.2 × 10 6 + 0.225 × 10 × 10 5 + 2.8 × 10 6 Rewrite 2.25 = 0.225 × 10 (4.2 + 0.225 + 2.8) × 10 6 7.225 × 10 6
Question 2. 8.5 × 10 3 − 5.3 × 10 3 − 1.0 × 10 2 8.5 × 10 3 − 5.3 × 10 3 − ? × 10? ? − ? − ? ? × 10 ? Type below: ______________
Answer: 8.5 × 10 3 − 5.3 × 10 3 − 0.1 × 10 3 (8.5 − 5.3 − 0.1) × 10 3 (3.1) × 10 3
Question 3. 1.25 × 10 2 + 0.50 × 10 2 + 3.25 × 10 2 Type below: ______________
Answer: 1.25 × 10 2 + 0.50 × 10 2 + 3.25 × 10 2 (1.25 + 0.50 + 3.25) × 10 2 5 × 10 2
Question 4. 6.2 × 10 5 − 2.6 × 10 4 − 1.9 × 10 2 Type below: ______________
Answer: 6.2 × 10 5 − 2.6 × 10 4 − 1.9 × 10 2 6.2 × 10 5 − 0.26 × 10 5 − 0.0019 × 10 5 (6.2 – 0.26 – 0.0019) × 10 5 5.9381 × 10 5
Multiply or divide. Write your answer in scientific notation.
Question 5. (1.8 × 10 9 )(6.7 × 10 12 ) Type below: ______________
Answer: 12.06 × 10 21
Explanation: (1.8 × 10 9 )(6.7 × 10 12 ) 1.8 × 6.7 = 12.06 10 9+12 = 10 21 12.06 × 10 21
Question 6. \(\frac { { 3.46×10 }^{ 17 } }{ { 2×10 }^{ 9 } } \) Type below: ______________
Answer: 1.73 × 10 8
Explanation: 3.46/2 = 1.73 10 17 /10 9 = 10 17-9 = 10 8 1.73 × 10 8
Question 7. (5 × 10 12 )(3.38 × 10 6 ) Type below: ______________
Answer: 16.9 × 10 18
Explanation: (5 × 10 12 )(3.38 × 10 6 ) 5 × 3.38 = 16.9 10 6+12 = 10 18 16.9 × 10 18
Question 8. \(\frac { { 8.4×10 }^{ 21 } }{ { 4.2×10 }^{ 14 } } \) Type below: ______________
Answer: 2 × 10 7
Explanation: 8.4/4.2 = 2 10 21 /10 14 = 10 21-14 = 10 7 2 × 10 7
Write each number using calculator notation.
Question 9. 3.6 × 10 11 Type below: ______________
Answer: 3.6e11
Question 10. 7.25 × 10 -5 Type below: ______________
Answer: 7.25e-5
Question 11. 8 × 10 -1 Type below: ______________
Answer: 8e-1
Write each number using scientific notation.
Question 12. 7.6E − 4 Type below: ______________
Answer: 7.6 × 10 -4
Question 13. 1.2E16 Type below: ______________
Answer: 1.2 × 10 16
Question 14. 9E1 Type below: ______________
Answer: 9 × 10 1
Question 15. How do you add, subtract, multiply, and divide numbers written in scientific notation? Type below: ______________
Answer: Numbers with exponents can be added and subtracted only when they have the same base and exponent. To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients, and subtract their exponents.
Question 16. An adult blue whale can eat 4.0 × 10 7 krill in a day. At that rate, how many krill can an adult blue whale eat in 3.65 × 10 2 days? Type below: ______________
Answer: 14.6 × 10 9
Explanation: (4.0 × 10 7 )(3.65 × 10 2 ) 4.0 × 3.65 = 14.6 10 7+2 = 10 9 14.6 × 10 9
Question 17. A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 × 10 13 cells. How many times as many cells does an adult have than a newborn? Write your answer in scientific notation. Type below: ______________
Answer: 1.9 × 10 3
Explanation: 26,000,000,000 = 2.6 × 10 10 4.94 × 10 13 (4.94 × 10 13 )/(2.6 × 10 10 ) 1.9 × 10 3
Represent Real-World Problems
Question 18. What is the total amount of paper, glass, and plastic waste generated? Type below: ______________
Answer: 11.388 × 10 7
Explanation: 7.131 × 10 7 + 1.153 × 10 7 + 3.104 × 10 7 11.388 × 10 7
Question 19. What is the total amount of paper, glass, and plastic waste recovered? Type below: ______________
Answer: 5.025 × 10 7
Explanation: 4.457 × 10 7 + 0.313 × 10 7 + 0.255 × 10 7 5.025 × 10 7
Question 20. What is the total amount of paper, glass, and plastic waste not recovered? Type below: ______________
Answer: 6.363 × 10 7
Explanation: (11.388 × 10 7 ) – (5.025 × 10 7 ) 6.363 × 10 7
Question 21. Which type of waste has the lowest recovery ratio? Type below: ______________
Answer: Plastics
Explanation: 7.131 × 10 7 – 4.457 × 10 7 = 2.674 × 10 7 1.153 × 10 7 – 0.313 × 10 7 = 0.84 × 10 7 3.104 × 10 7 – 0.255 × 10 7 = 2.849 × 10 7 Plastics has the lowest recovery ratio
Social Studies
Question 22. How many more people live in France than in Australia? Type below: ______________
Answer: 4.33 × 10 7
Explanation: (6.48 × 10 7 ) – (2.15× 10 7 ) 4.33 × 10 7
Question 23. The area of Australia is 2.95 × 10 6 square miles. What is the approximate average number of people per square mile in Australia? Type below: ______________
Answer: About 7 people per square mile
Explanation: 2.95 × 10 6 square miles = (2.15× 10 7 ) 1 square mile = (2.15× 10 7 )/(2.95 × 10 6 ) = 7.288
Question 24. How many times greater is the population of China than the population of France? Write your answer in standard notation. Type below: ______________
Answer: 20.52; there are about 20 people in china for every 1 person in France.
Question 25. Mia is 7.01568 × 10 6 minutes old. Convert her age to more appropriate units using years, months, and days. Assume each month to have 30.5 days. Type below: ______________
Answer: 13 years 3 months 22.5 days
Explanation: 7.01568 × 10 6 minutes (7.01568 × 10 6 minutes) ÷ (6 × 10 1 )(2.4 × 10 1 )(1.2 × 10 1 )(3.05 × 10 1 ) = (1.331 × 10 1 ) = 13 years 3 months 22.5 days
Question 26. Courtney takes 2.4 × 10 4 steps during her a long-distance run. Each step covers an average of 810 mm. What total distance (in mm) did Courtney cover during her run? Write your answer in scientific notation. Then convert the distance to the more appropriate unit kilometers. Write that answer in standard form. ______ km
Answer: 19.4 km
Explanation: Courtney takes 2.4 × 10 4 steps during her a long-distance run. Each step covers an average of 810 mm. (2.4 × 10 4 steps) × 810mm (2.4 × 10 4 ) × (8.1 × 10 2 ) The total distance covered = (19.44 × 10 6 ) Convert to unit kilometers: (19.44 × 10 6 ) × (1 × 10 -6 ) (1.94 × 10 1 ) 19.4 km
Question 27. Social Studies The U.S. public debt as of October 2010 was $9.06 × 10 12 . What was the average U.S. public debt per American if the population in 2010 was 3.08 × 10 8 people? $ _______
Answer: $29,400 per American
Explanation: ($9.06 × 10 12 .)/(3.08 × 10 8 ) ($2.94 × 10 4 .) = $29,400 per American
Question 28. Communicate Mathematical Ideas How is multiplying and dividing numbers in scientific notation different from adding and subtracting numbers in scientific notation? Type below: ______________
Answer: When you multiply or divide in scientific notation, you just add or subtract the exponents. When you add or subtract in scientific notation, you have to make the exponents the same before you can do anything else.
Question 29. Explain the Error A student found the product of 8 × 10 6 and 5 × 10 9 to be 4 × 10 15 . What is the error? What is the correct product? Type below: ______________
Answer: The error student makes is he multiply the terms instead of addition.
Explanation: product of 8 × 10 6 and 5 × 10 9 40 × 10 15 4 × 10 16 The student missed the 10 while multiplying the product of 8 × 10 6 and 5 × 10 9
Question 30. Communicate Mathematical Ideas Describe a procedure that can be used to simplify \(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \). Write the expression in scientific notation in simplified form. Type below: ______________
Answer: \(\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) } \) \(\frac { { (487×10 }^{ 10 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (61×10 }^{ 7 }) } \) (480 × 10 10 )/(64 × 10 7 ) 7.50 × 10³
Question 1. 3 -4 \(\frac{□}{□}\)
Answer: \(\frac{1}{81}\)
Explanation: Base = 3 Exponent = 4 3 -4 = (1/3) 4 = 1/81
Question 2. 35 0 ______
Explanation: 35 0 Base = 35 Exponent = 0 Anything raised to the zeroth power is 1. 35 0 = 1
Question 3. 4 4 ______
Answer: 256
Explanation: Base = 4 Exponent = 4 4 4 = 4 . 4 . 4 . 4 = 2561
Use the properties of exponents to write an equivalent expression.
Question 4. 8 3 ⋅ 8 7 Type below: ____________
Answer: 8 10
Explanation: 8 3 ⋅ 8 7 8 3+7 8 10
Question 5. \(\frac { 12^{ 6 } }{ 12^{ 2 } } \) Type below: ____________
Answer: 12 4
Explanation: 12 6 ÷ 12 2 12 6-2 12 4
Question 6. (10 3 ) 5 Type below: ____________
Answer: 10 8
Explanation: (10 3 ) 5 (10 3+5 ) (10 8 )
2.2 Scientific Notation with Positive Powers of 10
Convert each number to scientific notation or standard notation.
Question 7. 2,000 Type below: ____________
Answer: 2 × (10 3 )
Explanation: 2 × 1,000 Move the decimal left 3 places 2 × (10 3 )
Question 8. 91,007,500 Type below: ____________
Answer: 9.10075 × (10 7 )
Explanation: 91,007,500 Move the decimal left 7 places 9.10075 × (10 7 )
Question 9. 1.0395 × 10 9 Type below: ____________
Answer: 1039500000
Explanation: 1.0395 × 10 9 Move the decimal right 9 places 1039500000
Question 10. 4 × 10 2 Type below: ____________
Answer: 400
Explanation: 4 × 10 2 Move the decimal right 2 places 400
2.3 Scientific Notation with Negative Powers of 10
Question 11. 0.02 Type below: ____________
Answer: 2 × 10 -2
Explanation: 0.02 Move the decimal right 2 places 2 × 10 -2
Question 12. 0.000701 Type below: ____________
Answer: 7.01 × 10 -4
Explanation: 0.000701 Move the decimal right 4 places 7.01 × 10 -4
Question 13. 8.9 × 10 -5 Type below: ____________
Answer: 0.000089
Explanation: 8.9 × 10 -5 Move the decimal left 5 places 0.000089
Question 14. 4.41 × 10 -2 Type below: ____________
Answer: 0.0441
Explanation: 4.41 × 10 -2 Move the decimal left 2 places 0.0441
2.4 Operations with Scientific Notation
Perform the operation. Write your answer in scientific notation.
Question 15. 7 × 10 6 − 5.3 × 10 6 Type below: ____________
Answer: 1.7 × 10 6
Explanation: 7 × 10 6 − 5.3 × 10 6 (7 – 5.3) × 10 6 1.7 × 10 6
Question 16. 3.4 × 10 4 + 7.1 × 10 5 Type below: ____________
Answer: 7.44 × 10 4
Explanation: 3.4 × 10 4 + 7.1 × 10 5 0.34 × 10 5 + 7.1 × 10 5 (0.34 + 7.1) × 10 5 7.44 × 10 5
Question 17. (2 × 10 4 )(5.4 × 10 6 ) Type below: ____________
Answer: 10.8 × 10 10
Explanation: (2 × 10 4 )(5.4 × 10 6 ) (2 × 5.4)(10 4 × 10 6 ) 10.8 × 10 10
Question 18. \(\frac { 7.86×10^{ 9 } }{ 3×10^{ 4 } } \) Type below: ____________
Answer: 2.62 × 10 5
Explanation: 7.86/3 = 2.62 10 9 /10 4 = 10 5 2.62 × 10 5
Question 19. Neptune’s average distance from the Sun is 4.503×10 9 km. Mercury’s average distance from the Sun is 5.791 × 10 7 km. About how many times farther from the Sun is Neptune than Mercury? Write your answer in scientific notation. Type below: ____________
Answer: (0.7776 × 10 2 km) = 77.76 times
Explanation: As Neptune’s average distance from the sun is 4.503×10 9 km and Mercury is 5.791 × 10 7 km (4.503×10 9 km)/(5.791 × 10 7 km) (0.7776 × 10 9-7 km) (0.7776 × 10 2 km) 77.76 times
Essential Question
Question 20. How is scientific notation used in the real world? Type below: ____________
Answer: Scientific notation is used to write very large or very small numbers using less digits.
Question 1. Which of the following is equivalent to 6 -3 ? Options: a. 216 b. \(\frac{1}{216}\) c. −\(\frac{1}{216}\) d. -216
Answer: b. \(\frac{1}{216}\)
Explanation: Base = 6 Exponent = 3 6 3 = (1/6) 3 = 1/216
Question 2. About 786,700,000 passengers traveled by plane in the United States in 2010. What is this number written in scientific notation? Options: a. 7,867 × 10 5 passengers b. 7.867 × 10 2 passengers c. 7.867 × 10 8 passengers d. 7.867 × 10 9 passengers
Answer: c. 7.867 × 10 8 passengers
Explanation: 786,700,000 Move the decimal left 8 places 7.867 × 10 8 passengers
Question 3. In 2011, the population of Mali was about 1.584 × 10 7 people. What is this number written in standard notation? Options: a. 1.584 people b. 1,584 people c. 15,840,000 people d. 158,400,000 people
Answer: c. 15,840,000 people
Explanation: 1.584 × 10 7 Move the decimal right 7 places 15,840,000 people
Question 4. The square root of a number is between 7 and 8. Which could be the number? Options: a. 72 b. 83 c. 51 d. 66
Answer: c. 51
Explanation: 7²= 49 8²=64 (49+64)/2 56.5
Question 5. Each entry-level account executive in a large company makes an annual salary of $3.48 × 10 4 . If there are 5.2 × 10 2 account executives in the company, how much do they make in all? Options: a. $6.69 × 10 1 b. $3.428 × 10 4 c. $3.532 × 10 4 d. $1.8096 × 10 7
Answer: d. $1.8096 × 10 7
Explanation: Each entry-level account executive in a large company makes an annual salary of $3.48 × 10 4 . If there are 5.2 × 10 2 account executives in the company, ($3.48 × 10 4 )( 5.2 × 10 2 ) $1.8096 × 10 7
Question 6. Place the numbers in order from least to greatest. 0.24,4 × 10 -2 , 0.042, 2 × 10 -4 , 0.004 Options: a. 2 × 10 -4 , 4 × 10 -2 , 0.004, 0.042, 0.24 b. 0.004, 2 × 10 -4 , 0.042, 4 × 10 -2 , 0.24 c. 0.004, 2 × 10 -4 , 4 × 10 -2 , 0.042, 0.24 d. 2 × 10 -4 , 0.004, 4 × 10 -2 , 0.042, 0.24
Answer: d. 2 × 10 -4 , 0.004, 4 × 10 -2 , 0.042, 0.24
Explanation: 2 × 10 -4 = 0.0002 4 × 10 -2 = 0.04
Question 7. Guillermo is 5 \(\frac{5}{6}\) feet tall. What is this number of feet written as a decimal? Options: a. 5.7 feet b. 5.\(\bar{7}\) feet c. 5.83 feet d. 5.8\(\bar{3}\) feet
Answer: c. 5.83 feet
Question 8. A human hair has a width of about 6.5 × 10 -5 meters. What is this width written in standard notation? Options: a. 0.00000065 meter b. 0.0000065 meter c. 0.000065 meter d. 0.00065 meter
Answer: c. 0.000065 meter
Explanation: 6.5 × 10 -5 meter = 0.000065
Question 9. Consider the following numbers: 7000, 700, 70, 0.7, 0.07, 0.007 a. Write the numbers in scientific notation. Type below: _____________
Answer: 7000 = 7 × 10³ 700 = 7 × 10² 70 = 7 × 10¹ 0.7 = 7 × 10¯¹ 0.07 = 7 × 10¯² 0.007 = 7 × 10¯³
Question 9. b. Look for a pattern in the given list and the list in scientific notation. Which numbers are missing from the lists? Type below: _____________
Answer: In the given list the decimal is moving to the left by one place. From the scientific notation, numbers are decreasing by 10. The number missing is 7
Question 9. c. Make a conjecture about the missing numbers. Type below: _____________
Answer: The numbers will continue to decrease by 10 in the given list.
We wish the information provided in the Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation for all the students. Go through the solved examples to have a complete grip on the subject and also on the way of solving each problem. Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Key will help the students to score the highest marks in the exam.
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A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10.
When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.
Using Scientific Notation. Recall at the beginning of the section that we found the number 1.3 × 10 13 1.3 × 10 13 when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m.
Find step-by-step solutions and answers to Holt Algebra 1: Student Edition - 9780030358272, as well as thousands of textbooks so you can move forward with confidence. ... Powers and Exponents. Section 1-5: Square Roots and Real Numbers. Section 1-6: Order of Operations. Section 1-7: ... Powers of 10 and Scientific Notation. Section 7-3 ...
These Algebra 1 - Exponents Worksheets produces problems for working with Exponents with Multiplication. You may select the problems to contain only positive, negative or a mixture of different exponents. These Exponents Worksheets are a good resource for students in the 5th Grade through the 8th Grade. Exponents with Division Worksheets.
Scientific Notation A number is expressed in scientific notation when it is of the form \[a\space\times\space10^n \text{ where }1\leq a<10\text{ and } n \text{ is an integer.} \nonumber \] How to convert a decimal to scientific notation. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Unit: Exponents and Scientific Notation Cut and Paste Name Date PROPERTIES OF EXPONENTS CUT AND PASTE The expressions below have not been simplified. Cut out the cards of simplified expressions, and glue each simplified expression next to its match. EXPRESSION a 15 x (a6)-5 a 13 xa 33 a SIMPLIFIED ©Maneuvering the Middle LLC, 2015 EXPRESSION a- 22
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10.
Multiplying and Dividing Scientific Notation. à à Multiplying: Multiply the coefficients, add the exponents, and simplify to proper scientific notation. Example: 4×10 7×10 = 28×10 = 2.8×10. à à Dividing: Divide the coefficients, subtract the exponents, and simplify to proper scientific notation.
Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator's display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided 1 1 by 2,598,960 2,598,960 and saw the answer 3.848 × 10 −7 . 3.848 × 10 −7 .
Chart Maker. Geometry Calculator. Algebra Solver. Math Games. Calculator. Free worksheets (pdf) and answer keys on scientific notation. Each sheet is scaffolder and has model problems explained step by step.
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10.
Enduring Understandings (Know these concepts to do well on the test!) The Laws of Exponents make sense of very large & very small numbers. . Scientific Notation is a compact way of writing numbers that are very large or very small. Vocabulary (Make flash cards) Exponential notation. .
Students often move constants along with a variable that has a negative exponent. For example, in 4d) a common answer is 2a−3 = 1/(2a3). Refer students to the exponent rule charts and the Writing a Number in Scientific Notation chart in the text. Answers: 1a) 128; b) m17; c) −36xy2; d) 15a6b3; 2a) 1; b) −1; c) 1; d) 1; 3a) x5; b) −5y4 ...
To write a number less than 1 in scientific notation, move the decimal point right and use a negative exponent. When the number is The decimal point between 0 and 1, use 0.0 7 8 3 7.83 10 -2 moves 2 places to the a negative exponent. = × right. The average size of an atom is about 0.00000003 centimeter across.
Each pdf worksheet contains 14 problems rewriting whole numbers to scientific notation. Easy level has whole numbers up to 5-digits; Moderate level has more than 5-digit numbers. Students of 6th grade need to express each scientific notation in standard notation. An example is provided in each worksheet.
Exponents and scientific notation can fall into this trap. Vertical Alignment. Exponent Tips. It is important with both exponents and scientific notation that students understand that they show a different way to represent a value. Before even showing an exponent, start by showing the expanded form like 7*7*7*7*7. You can start by asking students:
HOW TO: Convert from decimal notation to scientific notation. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, n, that the decimal point was moved. Step 3. Write the number as a product with a power of 10.
TOS. Chart Maker. Geometry Calculator. Algebra Solver. Math Games. Calculator. Free worksheet with answer keys on exponents. Each one has model problems worked out step by step, practice problems, challenge proglems and youtube videos that explain each topic.
EXPONENTS AND SCIENTIFIC NOTATION UNIT TWO: ANSWER KEY ©MANEUVERING THE MIDDLE, 2016 Unit: Exponents and Scientific Notation Review .2 Name Date EXPONENTS AND SCIENTIFIC NOTATION STUDY GVIDC Solve each of the problems be ow. Be sure to ask questions if you need more help with a topic. 1 CAN APPLY EXPONENTS.
Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Answer Key. ... Diana calculated that she spent about 5.4 × 10 4 seconds doing her math homework during October. 5.4 × 10 4 Move the decimal right 4 places 5400. Question 14. The town recycled 7.6 × 10 6 cans this year. Write the number of cans in standard notation
We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation! When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product \((7\times{10}^4)⋅(5\times{10}^6)=35\times{10}^{10}\).