1. Scientific Notation
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Chapter 5
1.
Scientific Notation
Scientific notation is a powerful tool for representing very large or very small numbers in a more manageable form. The lesson delves into the mechanics of using exponents to simplify these numbers. It also highlights real-world applications, such as calculating distances in astronomy or understanding the scale of microscopic organisms. This method is not just a mathematical convenience; it is a crucial skill for fields like science, engineering, and finance. By mastering scientific notation, one can make more accurate calculations and better interpret data in various contexts.
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Lesson Settings & Tools
| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Scientific Notation
Slide of 16
Scientists and astronomers make calculations using huge and tiny numbers. They could go all the way up to 100 000 000 000 or as low as 0.00 000 000 001. Writing these numbers can take a lot of time while making calculations. A more practical way of writing numbers exists called scientific notation. This concept will be explored in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
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Distance From the Earth to the Sun
The distance from the Earth to the Sun is about 150 000 000 kilometers. 
An astronomer wants to find out how long a spaceship would take to fly to the Sun. He will need to write the given distance several times to make this calculation. Is there a way to make the given number shorter? If yes, how?
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Scientific Notation
Scientific notation is a compact way of writing very large or very small numbers. A number written in scientific notation is expressed as a product of two numbers. a * 10^b In this form, the first factor is greater than or equal to 1 and less than 10. In other words, it needs to be in the interval [1,10). The second factor is a power of 10 where b is an integer. For example, the number 4 million can be rewritten as the product of 4 and a multiple of 10. Then, the multiple of 10 is rewritten as a base 10 power. 4 000 000 = 4 * 1 000 000 = 4 * 10^6 Very small decimal numbers can also be written in scientific notation. Consider a number where there are many zeros before the significant figures. Take as 0.000342 as an example. 0.000342 = 3.42 ÷ 10 000 = 3.42 * 10^(- 4) In such cases, numbers are expressed as a division by a multiple of 10. Division by a multiple of 10 is equivalent to multiplication by a base 10 power with a negative exponent. Consider a few more examples of numbers written in scientific notation.
| Decimal Form | Written as a Product or Division Expression | Scientific Notation |
|---|---|---|
| 4505 | 4.505 * 1000 | 4.505 * 10^3 |
| 8 320 000 | 8.32 * 1 000 000 | 8.32 * 10^6 |
| 0.0005 | 5 ÷ 10 000 | 5 * 10^(-4) |
| 0.0521 | 5.21 ÷ 100 | 5.21 * 10^(-2) |
Intuitive Method: Rewriting a Number in Scientific Notation
An intuitive method to rewrite a number into scientific notation is to count the number of places the decimal needs tomove. Consider a number greater than 10. The decimal would move from right to left to make the number less than 10 but still greater than 1. The number of places the decimal moved indicates the positive exponent to be used for the base 10 power.
Example
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Rewriting the Heights of Mountains
Ramsha and her friends did a research project about the highest mountains in the world. They found that Mount Everest is the world's highest mountain. It is located in the Himalayan mountain range. They also learned that Mount Elbrus is the highest peak of Europe. It is in southwestern Russia.
a The peak of Mount Everest is at 8 848 meters above the sea level. Rewrite this height in scientific notation.
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b Mount Elbrus is an extinct volcano with twin cones that reach 1.851 * 10^4 feet. Rewrite this height in standard form.
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Solution
a The height of Mount Everest is 8 848 meters. Notice that this number is four digits long when written in standard form. Start by placing the decimal point after the first non-zero digit to write it in scientific notation.
8. 848 ↑ First nonzero digit Next, count the digits after the decimal point. 8. 848 ↑ Digits after the decimal point There are three digits after the decimal point. This number will be written as the exponent of 10. Standard Form & Scientific Notation 8 848 & 8. 848 * 10^3
b This time the number is given in the scientific notation and the task is to rewrite it into standard form.
1.851 * 10^4 The power of 10 is 4. It has a positive exponent. Therefore, the decimal point will be moved to the right 4 times. In other words, the first factor 1.851 will be multiplied by 10 four times.
The diagram shows that the decimal point continues to move after the nonzero digits. Zeroes can be added to the end of the number until the move is done. Note that adding zeroes to the end of a decimal number after the decimal point does not change the value of the number. 1.851=1.8510000 ... In this case, there are already three nonzero digits. Only one zero needs to be added to the end of the number. Now, the number is in standard form as a five-digit number. Scientific Notation & Standard Form 1.851 * 10^4 & 18 510
Example
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Rewriting the Lengths of Bacteria
Ramsha and her friends continue to enjoy their school studies. Next is biology class! They use a microscope to examine two types of bacteria, Escherichia coli and Salmonella.
a The length of an E. coli bacterium is two micrometers long. That is equal to 0.000002 meters. Write this length in scientific notation in meters.
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b The length of a salmonella bacterium is 1.5 * 10^(-6) meters. Rewrite this length in standard form.
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Hint
a Start by placing the decimal point after the first non-zero digit.
Solution
a The length of an Escherichia coli bacterium is given as 0.000002 meters. The goal is to write this number in scientific notation. Begin by placing the decimal point after the first non-zero digit.
0 . 0 0 0 0 0 2. ↑ First nonzero digit Then, determine the power of 10. That is done by counting the number of digits before the new decimal point. 0. 0 0 0 0 0 2. ↑ Digits before the new decimal point There are six digits before the decimal point. The given number 0.000002 is less than 1. That means the exponent will be negative. Standard Form & Scientific Notation 0. 000002 & 2 * 10^(-6)
b The length of a salmonella bacterium is given in scientific notation.
1.5 * 10^(-6) Notice that the power of 10 is negative. This means that the decimal point will be moved to the left six times.
A zero is added every time the decimal moves to the left of the given digits. Remember to write an additional zero before the decimal point. Finally, the number is rewritten in standard form as an eight-digit number. Scientific Notation & Standard Form 1.5 * 10^(-6) & 0.0000015
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Translating Between Scientific Notation and Standard Form
Rewrite the given expression in standard form if it is given in scientific notation. Or rewrite it in scientific notation if it is given in standard form.
Example
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Examining the Populations of Some Countries
Ramsha's school has an interactive electronic map that shows population data around the world! Ramsha's geography teacher asked her class to examine the populations of some countries.
They used this map to see the numbers of people living in Brazil, Turkey, China, Australia, and the United States of America.
a Write the number of people living in these countries in scientific notation by rounding the given numbers to the greatest place value.
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b Sort the countries from greatest to least population.
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Solution
a The populations of the five countries will be written in scientific notation one at a time. Start by rounding all the numbers to the greatest place value in a table.
| Population | Rounded | |
|---|---|---|
| USA | 331 002 651 | 300 000 000 |
| Brazil | 212 559 417 | 200 000 000 |
| Turkey | 84 339 067 | 80 000 000 |
| China | 1 439 323 776 | 1 000 000 000 |
| Australia | 25 499 884 | 30 000 000 |
Now all the rounded numbers can be rewritten as a single digit times a power of 10. Count the zeros to determine the power of 10 for each number. For instance, the rounded population of the USA can be rewritten in this way. 3 00 000 000 = 3 * 10^8 Apply the same method so the rounded populations of other countries in standard form can be rewritten in scientific notation.
| Population | Rounded | Scientific Notation | |
|---|---|---|---|
| USA | 331 002 651 | 3 00 000 000 | 3* 10^8 |
| Brazil | 212 559 417 | 2 00 000 000 | 2* 10^() 8 |
| Turkey | 84 339 067 | 8 0 000 000 | 8* 10^() 7 |
| China | 1 439 323 776 | 1 000 000 000 | 1* 10^() 9 |
| Australia | 25 499 884 | 3 0 000 000 | 3* 10^() 7 |
b It is time to determine which country has the greatest population! This process requires comparing the numbers written in scientific notation from Part A.
| Country | Population |
|---|---|
| USA | 3* 10^() 8 |
| Brazil | 2* 10^() 8 |
| Turkey | 8* 10^() 7 |
| China | 1* 10^() 9 |
| Australia | 3* 10^() 7 |
Examine the powers of 10. Begin with identifying the largest power. China has the largest. That means it has the greatest population. Now compare the numbers with the same power of 10. Start with 10^8 because 10^8 is greater than 10^7. USA & Brazil 3* 10^() 8 & 2* 10^() 8 The powers of 10 for the US and Brazil are equal. That means their first factors should be compared. The value 3 is larger than 2. That means the population of the USA is greater than Brazil. Next, compare the populations of Turkey and Australia. Turkey & Australia 8* 10^() 7 & 3* 10^() 7 Again, the powers of 10 are the same. This indicates that the values of 8 and 3 should be checked. Well, 8 is greater than 3. That means the population of Turkey is greater than Australia. Now the countries can be sorted from greatest to least population. c China & & 1* 10^() 9 USA & & 3* 10^() 8 Brazil & & 2* 10^() 8 Turkey & & 8* 10^() 7 Australia & & 3* 10^() 7
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Multiplying Numbers in Scientific Notation
Numbers written in scientific notation can be multiplied by using the properties of exponents. Two numbers written in scientific notation a* 10^b and c * 10^d can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.
( a* 10^b) * ( c* 10^d)= a c * 10^(b+d)
1
Write Each Number in Scientific Notation
expand_more It is helpful to check whether both of the numbers are written in scientific notation. Recall that a number is in scientific notation if it is written as a product of two numbers. One of these numbers should be a power of 10. The other number should be greater than or equal to 1 and less than 10. c First Factor & * & Second Factor 1.5 & * & 10^2 & ✓ 12 & * & 10^5 & * The first number is in scientific notation. The second number is not. Its first factor 12 is greater than 10. Note that 12 can also be expressed as 12.0. The decimal point needs to move one unit to the left for the first factor to become less than 10. The second factor's power of 10 is then increased by that number of moves, 1. 12 * 10^5 ⇔ 1.2 * 10^(5+1) ⇔ 1.2*10^6 ✓
2
Multiply the First Factors
expand_more Both numbers are now written in scientific notation. The first factors of both numbers can be multiplied by using the Commutative Property of Multiplication.
(1.5 * 10^2)*(1.2* 10^6)
RemovePar
Remove parentheses
1.5 * 10^2 * 1.2* 10^6
CommutativePropMult
Commutative Property of Multiplication
1.5 * 1.2 * 10^2 * 10^6
Multiply
Multiply
1.8 * 10^2 * 10^6
3
Multiply the Second Factors
expand_more Next, the powers with base 10 are multiplied. Remember to use the Product of Powers Property — exponents are added when multiplying powers with the same base.
4
Write the Result in Scientific Notation
expand_more The result is 1.8 * 10^8. The first factor is greater than 1 and less than 10. The second factor is written as a power of 10. Therefore, the result is already in scientific notation.
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Dividing Numbers in Scientific Notation
Numbers written in scientific notation can be divided by using the properties of exponents. Use the Quotient of Powers Property to divide numbers written in scientific notation.
a* 10^b/c* 10^d = a/c * 10^(b-d)
1
Write Each Number in Scientific Notation
expand_more Start by checking whether each number is written in scientific notation. c First Factor & * & Second Factor 0.36 & * & 10^(23) & * The first factor is less than 1. That means the dividend still needs to be written in scientific notation. Move the decimal point to the right until 0.36 is greater than 1 and less than 10. Then, decrease the second factor's exponent by the number of moves. 0.36 * 10^(23) ⇔ 3.6 * 10^(22) Next, consider the divisor. The divisor 1200 is a whole number. Note that 1200 can also be expressed as 1200.0. Now, move the decimal point three units to the left to get a first factor that is less than 10 and greater than 1. The second factor can then be written as 10 to the power of 3. 1200 ⇔ 1.2 * 10^3
2
Divide the First Factors
expand_more Now that both the dividend and divisor are written in scientific notation, the first factors of both can be divided.
3.6 * 10^(22)/1.2 * 10^3
WriteProdFrac
Write as a product of fractions
3.6/1.2 * 10^(22)/10^3
CalcQuot
Calculate quotient
3 * 10^(22)/10^3
3
Divide the Second Factors
expand_more 4
Write the Result in Scientific Notation
expand_more Finally, write the resulting number in scientific notation. Notice that in this instance, the quotient is already in scientific notation. 3 * 10^(19)
Example
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Finding the Product and Quotient
Ramsha's math teacher introduced how to multiply and divide numbers written in scientific notation. She then asked the class to work on the following examples.
a Solve along with Ramsha to find the product given in Example I. Express the result in scientific notation.
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b Solve along with Ramsha to find the quotient given in Example II. Express the result in scientific notation.
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Hint
a Use the Commutative Property of Multiplication and the Product of Powers Property.
b Use the Quotient of Powers Property.
Solution
a Ramsha is finding the product of two numbers. It is important to check that both numbers are in scientific notation before she finds their product. She can do this by checking for two certain characteristics.
- The first factor must be greater than or equal to 1 and less than 10.
- The second factor must be a power of 10.
(1.4 * 10^(44)) * (5 * 10^(14))
RemovePar
Remove parentheses
1.4 * 10^(44) * 5 * 10^(14)
CommutativePropMult
Commutative Property of Multiplication
1.4 * 5 * 10^(44) * 10^(14)
Multiply
Multiply
7 * 10^(44) * 10^(14)
MultPow
a^m*a^n=a^(m+n)
7 * 10^(58)
1.4 * 10^(44)/5 * 10^(14)
WriteProdFrac
Write as a product of fractions
1.4/5 * 10^(44)/10^(14)
CalcQuot
Calculate quotient
0.28 * 10^(44)/10^(14)
DivPow
a^m/a^n= a^(m-n)
0.28 * 10^(30)
Example
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Snails on the Way
Ramsha loves reading science magazines. An article she read said that garden snails move at an incredibly slow speed of only 0.048 kilometers per hour!
a How many kilometers can a snail go if it is on the move for a full day of 24 hours? Perform the needed operations in scientific notation. Also express the result in scientific notation. Round the result to two decimal places if necessary.
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b How long does it take a snail to move one meter? Perform the needed operations in scientific notation. Also express the result in scientific notation. Round the result to two decimal places if necessary.
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Hint
a Distance traveled can be calculated by multiplying the speed by the time.
b Use the Quotient of Powers Property.
Solution
a It is given that a snail can move 0.048 kilometers in one hour. One day is 24 hours, so the distance traveled in one day can be found by multiplying the speed of the snail by 24.
( 4.8 * 10^(-2)) * ( 2.4 * 10^1)
CommutativePropMult
Commutative Property of Multiplication
(4.8 * 2.4) * ( 10^(-2) * 10^1)
Multiply
Multiply
11.52 * (10^(-2) * 10^1)
MultPow
a^m*a^n=a^(m+n)
11.52 * 10^(-1)
b This time it is asked for the time that it takes to move one meter for the snail. Since the speed of the snail is given in terms of kilometers per hour, at first rewrite one meter in terms of kilometers. There are 1000 meters in 1 kilometer, so 1 meter is one-thousands or 0.001 of a kilometer.
1 * 10^(-3)/4.8 * 10^(-2)
WriteAsFrac
Write as a fraction
1/4.8 * 10^(-3)/10^(-2)
UseCalc
Use a calculator
(0.208333 ...) * 10^(-3)/10^(-2)
RoundDec
Round to 3 decimal place(s)
0.208 * 10^(-3)/10^(-2)
DivPow
a^m/a^n= a^(m-n)
0.208 * 10^(-3-(-2))
SubNeg
a-(- b)=a+b
0.208 * 10^(-3+2)
AddTerms
Add terms
0.208 * 10^(-1)
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Multiply or Divide Numbers in Scientific Notation
Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.
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Add and Subtract Numbers in Scientific Notation
Numbers written in scientific notation can be added or subtracted by adding or subtracting the first factors if the powers of 10 are the same.
( a* 10^b) ± ( c* 10^b)=( a ± c) * 10^b
1
Change the Exponents of 10 to Be the Same
expand_more Two options can be considered. Either increase the power of 10^(12) or decrease the power of 10^(15). The second option will be considered here. Move the decimal point to the right three units. Then decrease the power of 10 by the same number of moves, three units. 0.15 * 10^(15) ⇔ 150 * 10^(12) Now, both of the numbers have the same 10-base factor with the exponent of 12.
2
Add or Subtract the First Factors
expand_more The obtained numbers are now like terms. That means the first factors are ready to be added.
3
Write the Result in Scientific Notation
expand_more Notice that the result 153 * 10^(12) is not in scientific notation. The first factor 153 is greater than 10. It needs to be less than 10 and greater than or equal to 1. That is done by moving the decimal point two units to the left. Then increase the power of 10 by two units. 153 * 10^(12) ⇔ 1.53 * 10^(14)
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Calculating the Energy Produced by Wind Turbines
Ramsha's class took a field trip to learn about wind turbines. The turbines supply lots of households with electricity. A typical large wind turbine can produce 6 * 10^6 kilowatt-hour energy per year. A small wind turbine can produce 1.3 * 10^5 kilowatt-hour energy per year.
a How much energy can a typical large and small wind turbine produce together in one year? Write the result in scientific notation.
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b What is the difference in the produced energies of the two wind turbines per year in scientific notation?
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Hint
a Rewrite the numbers to have the same power of 10. When the numbers are like terms, add the first factors of these numbers.
b Move the decimal point to rewrite the numbers. When the numbers are like terms, subtract the first factors of these numbers.
Solution
a The goal is to find the total energy that is produced by a typical large wind turbine and a small wind turbine together. The given energy amounts per year will be added to find that number.
b This time the difference between the energy amounts will be found.
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Add or Subtract Numbers in Scientific Notation
Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.
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Writing the Distance From the Earth to the Sun
The initial challenge of this collection stated that the distance from the Earth to the Sun is about 150 000 000 kilometers. 
An astronomer making calculations with this number needs to write it several times to find the time it would take a spaceship to fly to the Sun. Is there a way to make this notation shorter? If yes, how?
Answer
Yes, by using scientific notation.
Solution
This number can be written as a product of two numbers to express it in shorter notation. 150 000 000 = First Factor * Second Factor There are lots zeros at the end of the given number — seven zeros to be exact. These seven zeros can be written as a power of 10. The remaining two digits 1 and 5 can be written as the other factor. 15 0 000 000 ⇓ 15 * 10^7 Consider the main topic of this lesson, scientific notation. A number written in scientific notation needs to have the following two certain characteristics.
- The first factor must be greater than or equal to 1 and less than 10.
- The second factor must be a power of 10.
With this in mind, consider the obtained number in the quest to represent 150 000 000 kilometers. 15 * 10^7 The first factor is greater than 10. That means it needs to be rewritten once again. That can be done by moving the decimal point one unit to the left. This results in the power of 10 increasing by one unit. 15 * 10^7 ⇔ 1.5 * 10^8 Writing it this way means the astronomer spends less time writing out 150 000 000 kilometers multiple times. That might not sound like such a big deal, but over time it really makes a difference. Imagine writing such a large number over and over again. Thank you scientific notation!
Scientific Notation
Exercise 2.1
Scientists are performing an experiment about bacteria. They note that about 3.6 * 10^8 bacteria grew in 4 petri dishes in total.
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We know that about 3.6 * 10^8 bacteria are growing in 4 petri dishes in total. We are trying to find how many bacteria are growing in each petri dish. The number of bacteria in each petri dish is the same. That means we can divide the total number of bacteria growing in 4 petri dishes by 4. 3.6*10^8/4 Let's first check whether both the numerator and denominator are in scientific notation. First Factor & Second Factor 3.6 ✓ & 10^8 ✓ 4 ✓ & ? Note that 3.6*10^8 is already in scientific notation. The number of petri dishes is less than 10 and greater than 1. However, we should express it as a product of 4 and a power of 10. We can do that by multiplying it by 10^0. That value equals 1 according to the Identity Property of Multiplication. 3.6*10^8/4*10^0 Great! Now, we can calculate the quotient in the expression using the Quotient of Powers Property.
Note that the result is not in scientific notation because 0.9 is less than 1. Let's move the decimal point one place to the right and decrease the power of 10. 0.9 * 10^8 ⇔ 9 * 10^7 Reconsider context of the problem. We can say that there are 9* 10^7 bacteria in each petri dishes.
This time we will find the total number of bacteria in 20 petri dishes. We will multiply the number of bacteria growing in each petri dish by the number of petri dishes. We already found that there are 9* 10^7 bacteria in each petri dishes in the Part I of the exercise. ( 9 * 10^7 ) * 20 Now we will need to evaluate the obtained expression. Let's start by checking whether both numbers are in scientific notation. First Factor & Second Factor 9 ✓ & 10^7 ✓ 20 ✓ & ? The number of petri dishes is not less than 10. That means it is not written in scientific notation. We can express it in scientific notation by rewriting 20 as 2* 10^1. ( 9 * 10^7 ) * ( 2* 10^1 ) Next, we can calculate the product in the expression by using the Product of Powers Property.
The first factor of 18 is not less than 10. It is not written in scientific notation. We can express it in scientific notation by moving the decimal point one place to the left and increase the power of 10. 18 * 10^8 ⇔ 1.8 * 10^9 A total of 1.8* 10^9 bacteria are growing in 20 petri dishes.
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Sirius is the brightest star in Earth's night sky. It is about 8.6 light years from Earth. One light year is about 5.9 * 10^(12) miles. How far from Earth is Sirius measured in miles? Express the distance in scientific notation.
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The distances between Sirius and Earth is given in light years. The distance of one light year in miles is also given. The question asks us to give the distance between Sirius and Earth in miles. We need to convert the distance given in light years to miles.
Let's start by multiplying the distance in light years by the number of miles in a light year. Distance in miles = ( 8.6 ) * ( 5.9 * 10^(12) ) Now we want to evaluate the obtained expression using the Product of Powers Property. Let's do it!
Note that 50.74 is not less than 10. That means our result is not written in scientific notation, yet. We can express it in scientific notation by moving the decimal point one place to the left and increasing the power of 10. 50.74 * 10^(12) ⇔ 5.074 * 10^(13) The distance between Sirius and Earth is about 5.074 * 10^(13) miles.
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The mass of Earth is about 5.97* 10^(24) kilograms. The mass of the Moon is about 7.35* 10^(22) kilograms.
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We want to find the combined mass of Earth and the Moon expressed in scientific notation. Let's begin by adding the mass of Earth to the mass of the Moon. ( 5.97 * 10^(24) )+( 7.35* 10^(22) ) The masses should be written with the same power of 10. With this in mind, let's rewrite the first number. The decimal point should be moved two places to the right and the power of 10 decreased by two units. 5.97 * 10^(24) ⇔ 597 * 10^(22) Continue to focus on the first factors of the numbers. These numbers are now like terms. That means they can be added. Factoring out 10^(22) gets the process started.
Notice that the result is not in scientific notation because the first factor is greater than 10. It needs to be rewritten. The decimal point moves two places to the left and the power of 10 increases by two units. 604.35 * 10^(22) ⇔ 6.0435 * 10^(24) We found the combined mass of Earth and the Moon equals 6.0435 * 10^(24) kilograms.
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The greatest distance between the Sun and Earth is about 1.52 * 10^8 kilometers. The greatest distance between the Sun and Saturn is about 1.52 * 10^9 kilometers. Find the difference between these two distances. Write the result in scientific notation.
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We are asked to find the difference between two distances. Subtracting the lesser distance from the greater one will accomplish that. Writing down the distances using the same power of 10 helps us compare them. Begin by rewriting 1.52 * 10^9. The decimal point moves one place to the right and the power of 10 is decreased by one unit. 1.52 * 10^9 ⇔ 15.2 * 10^8 Now the second factors of the products are the same. We can now compare the greatest distance between the Sun and Earth with the greatest distance between the Sun and Saturn. 1.52 * 10^8 < 15.2 * 10^8 Let's express their difference as an expression. ( 15.2 * 10^8 ) - ( 1.52 * 10^8 ) The numbers are written with the same power of 10. That allows for us to factor out 10^8.
We end up with 13.68 which is greater than 10. The result is not written in scientific notation yet. Moving the decimal point one place to the left makes the first factor less than 10. That results in increasing the power of 10 by one unit. 13.68 * 10^8 ⇔ 1.368 * 10^9 The difference between the given two distances equals 1.368* 10^9 kilometers.
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