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.
Show less Show more expand_more| Student Learning Objectives: |
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| | 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.
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Challenge
Distance From the Earth to the Sun
The distance from the Earth to the Sun is about 150 000 000 kilometers.
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Discussion
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.
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Example
Rewriting the Heights of Mountains
Mount Everest is the world's highest mountain. It is located in the Himalayan mountain range. Mount Elbrus is the highest peak of Europe. It is in southwestern Russia.
a
The peak of Mount Everest is at 8848 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 8848 meters. Start by placing the decimal point after the first nonzero 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 3 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
We want to rewrite the given number in standard form.
1.851 * 10^4 The power of 10 is 4. The exponent is positive, so we will move the decimal point to the right 4 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. Here we only need to add one more zero to complete our multiplication. Scientific Notation & Standard Form 1.851 * 10^4 & 18 510
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Example
Rewriting the Lengths of Bacteria
Two bacteria that commonly cause food poisoning are Escherichia coli and Salmonella.
a
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
A salmonella bacterium is 1.5 * 10^(-6) meters long. 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 E. coli bacterium is given as 0.000002 meters. We want to write this number in scientific notation. Let's start by placing the decimal point after the first nonzero digit.
0 . 0 0 0 0 0 2. ↑ First nonzero digit Then we determine the power of 10 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 6 digits before the decimal point. The given number 0.000002 is less than 1, so 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) The power of 10 is negative, so we will move the decimal point left 6 times.
We add a zero every time the decimal moves to the left of the given digits. Remember to write an additional zero before the decimal point. Scientific Notation & Standard Form 1.5 * 10^(-6) & 0.0000015
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Pop Quiz
Translating Between Scientific Notation and Standard Form
If the given number is in scientific notation, rewrite it in standard form. If the given number is in standard form, rewrite it in scientific notation.
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Example
Examining the Populations of Some Countries
The table shows the populations of several countries in 2020.
| Country | Population (people) |
|---|---|
| Australia | 25 499 884 |
| Brazil | 212 559 417 |
| China | 1 439 323 776 |
| Turkey | 84 339 067 |
| USA | 331 002 651 |
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
We will write the populations of the five countries in scientific notation one at a time. Start by rounding all the numbers to the greatest place value in a table.
| Country | Population | Rounded |
|---|---|---|
| Australia | 25 499 884 | 30 000 000 |
| Brazil | 212 559 417 | 200 000 000 |
| China | 1 439 323 776 | 1 000 000 000 |
| Turkey | 84 339 067 | 80 000 000 |
| USA | 331 002 651 | 300 000 000 |
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. Let's rewrite the population for Australia. 3 0 000 000 = 3 * 10^7 Apply the same method to the rounded populations of other countries to rewrite them in scientific notation.
| Population | Rounded | Scientific Notation | |
|---|---|---|---|
| Australia | 25 499 884 | 3 0 000 000 | 3* 10^() 7 |
| Brazil | 212 559 417 | 2 00 000 000 | 2* 10^() 8 |
| China | 1 439 323 776 | 1 000 000 000 | 1* 10^() 9 |
| Turkey | 84 339 067 | 8 0 000 000 | 8* 10^() 7 |
| USA | 331 002 651 | 3 00 000 000 | 3* 10^8 |
b
Now we want to order the countries by their populations. We can do this more easily by using the scientific notations from Part A.
| Country | Population |
|---|---|
| Australia | 3* 10^() 7 |
| Brazil | 2* 10^() 8 |
| China | 1* 10^() 9 |
| Turkey | 8* 10^() 7 |
| USA | 3* 10^() 8 |
Let's find the the largest power of 10. China has the largest power, so it has the greatest population. Now let's compare the numbers with the same power of 10, starting with 10^8. USA & Brazil 3* 10^() 8 & 2* 10^() 8 The powers of 10 for the USA and Brazil are equal, so we compare their first factors. Since 3 is larger than 2, the population of the USA is greater than that of Brazil. Next, compare the populations of Turkey and Australia. Turkey & Australia 8* 10^() 7 & 3* 10^() 7 The powers of 10 are the same, so let's compare the first factors. We know that 8 is greater than 3, so the population of Turkey is greater than that of Australia. Now we can sort the countries 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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Discussion
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)
Consider the following product. (1.5 * 10^2)*(12 * 10^5) Followed four steps to multiply these numbers.
1
Write Each Number in Scientific Notation
expand_more It is helpful to check whether both of the numbers are written in scientific notation. c
First Factor & * & Second Factor 1.5 & * & 10^2 & ✓ 12 & * & 10^5 & *
The first number is in scientific notation, but the second number is not. Its first factor 12 is greater than 10. 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, multiply the powers with base 10 using the Product of Powers Property — exponents are added when multiplying powers with the same base.
4
Write the Result in Scientific Notation
expand_more Finally, write the resulting number in scientific notation if necessary. Our product is already in scientific notation.
1.8* 10^8
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Discussion
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)
Consider the division of the following two numbers. 0.36 * 10^(23)/1200 Numbers written in scientific notation can be divided in four steps.
1
Write Each Number in Scientific Notation
expand_more Start by checking whether each number is written in scientific notation. Let's start with the dividend.
c
First Factor & * & Second Factor 0.36 & * & 10^(23) & *
The first factor is less than 1, so rewrite the dividend in scientific notation. Move the decimal point to the right until the factor is at least 1 but less than 10. For each place the decimal point moves to the right, decrease the exponent by 1.
0.36 * 10^(23) ⇔ 3.6 * 10^(22)
The divisor 1200 can be written as 1200.0. Move the decimal point three places to the left to get 1.2. Since the decimal point moved three places, the second factor is 10^3.
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 if necessary. Our quotient is already in scientific notation.
3 * 10^(19)
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Example
Finding the Product and Quotient
a
Find the product. Express the result in scientific notation.
(1.4* 10^(44))* (5*10^(14))
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b
Find the quotient. Express the result in scientific notation.
1.4* 10^(44)/5* 10^(14)
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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
The first step to finding the product is to confirm that both numbers are in scientific notation. We 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.
Let's check our numbers. c First Factor & &Second Factor 1.4 & * & 10^(44) & ✓ 5 & * & 10^(14) & ✓ Both numbers meet the two characteristics, so they are written in scientific notation. Now we will multiply their first factors using the Commutative Property of Multiplication. Then we will multiply their second factors by using the Product of Powers Property.
(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)
The product of the multiplication is already in scientific notation!
b
Earlier we confirmed that both values are already in scientific notation. We will divide the first factors like fractions and the second factors by using Quotient of Powers Property.
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)
The result is not in scientific notation yet because the first factor is less than 1. Let's move the decimal 1 place to the right and decrease the power of 10 by 1. 0.28 * 10^(30) ⇔ 2.8 * 10^(29) Now the quotient is in scientific notation.
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Example
Snails on the Way
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 24 hours? 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 the snail to move one meter? 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
We are given that a snail can move 0.048 kilometers in one hour. We can find the distance the snail can travel in 24 hours by multiplying this speed by 24.
Distance = Speed * Time ⇓ Distance= 0.048 * 24 Begin by rewriting both numbers in scientific notation. Since 0.048 is less than 1, move the decimal point to get 4.8. Because the decimal point moved 2 places to the right, the exponent is -2. 0.048 * 10^0 = 4.8 * 10^(-2) To rewrite 24 in scientific notation, we move the decimal point 1 space to the left. Movement to the left means the exponent of the base 10 power will be positive. The second factor is 10^1. 24 * 10^0 = 2.4 * 10^1 Now that both numbers are in scientific notation, we multiply them by using Commutative Property of Multiplication and Product of Powers Property.
( 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)
We need to rewrite the product in scientific notation. Move the decimal point 1 unit to the left and increase the power of 10 by 1. 11.52 * 10^(-1) = 1.152 * 10^0 Finally, round the result to two decimals. 1.152 * 10^0 ≈ 1.15 * 10^0 In a full day, the snail can move about 1.15 * 10^0, or 1.15, kilometers.
b
We want to find the time that it takes the snail to move one meter. Since the speed of the snail is given in terms of kilometers per hour, we will at first rewrite one meter in terms of kilometers. There are 1000 meters in 1 kilometer, so 1 meter is one-thousandth, or 0.001, of a kilometer.
1 meter = 0. 001 kilometers Now we rearrange the distance formula to find time by dividing both sides of the equation by speed. Distance = Speed * Time ⇓ Time=Distance/Speed Before we use the formula, rewrite the numbers in scientific notation. 00.001 * 10^0 &=& 1 * 10^(-3) 0.048 * 10^0 &=& 4.8 * 10^(-2) Then, perform the division to find the time that the snail needs to move 1 * 10^(-3) kilometers.
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)
Let's rewrite this value in scientific notation. 0.208 * 10^(-1) = 2.08 * 10^(-2) This means that the snail moves one meter in about 2.08 * 10^(-2) hours. This is about 75 seconds. Turns out snails are actually not that slow!
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Pop Quiz
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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Discussion
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 and only if the powers of 10 are the same.
( a* 10^b) ± ( c* 10^b)=( a ± c) * 10^b
Consider the following addition example. (3 * 10^(12))+(0.15 * 10^(15)) Follow these three steps to add the numbers.
1
Change the Exponents of 10 to Be the Same
expand_more We can either increase the power of 10^(12) or decrease the power of 10^(15). We will decrease the power of 10^(15) for this solution. Move the decimal point 3 places to the right and decrease the power of 10 by 3.
0.15 * 10^(15) ⇔ 150 * 10^(12)
Now both numbers have 10^(12) as a factor.
2
Add or Subtract the First Factors
expand_more The numbers are now like terms. We can now add the first factors together.
(3 * 10^(12))+(150 * 10^(12))
FactorOut
Factor out 10^(12)
(3+150) * 10^(12)
AddTerms
Add terms
153 * 10^(12)
3
Write the Result in Scientific Notation
expand_more The sum is not in scientific notation. Move the decimal point 2 places to the left and increase the exponent of 10 by 2. 153 * 10^(12) ⇔ 1.53 * 10^(14)
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Example
Calculating the Energy Produced by Wind Turbines
A typical large wind turbine can produce 6 * 10^6 kilowatt-hours energy per year. A small wind turbine can produce 1.3 * 10^5 kilowatt-hours 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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Solution
a
We want to add the given numbers together.
(6 * 10^6) + (1.3 * 10^5) The powers of 10 need to be the same to be able to add these numbers. The first number can be rewritten by moving the decimal point 1 place to the right. Remember to decrease the power of 10 by 1. 6 * 10^6 ⇔ 60 * 10^5 Now that the numbers are like terms, we can add the first factors together.
The sum is not in scientific notation because the first factor is greater than 10. Let's rewrite it. 61.3 * 10^5 ⇔ 6.13 * 10^6 In total, a typical large and small wind turbine produce 6.13 * 10^6 kilowatt-hours energy per year. That is enough to meet the electricity demand of around 1600 average households per year.
b
We want to find the difference between the energy amounts.
(6 * 10^6) - (1.3 * 10^5) In Part A, we rewrote the first number so that the values become like terms. ccc ( 6 * 10^6) & - & (1.3 * 10^5) & ⇓ & ( 60 * 10^5) & - & (1.3 * 10^5) Now we can subtract them.
(60 * 10^5) - (1.3 * 10^5)
FactorOut
Factor out 10^5
(60-1.3) * 10^5
SubTerms
Subtract terms
58.7 * 10^5
Let's give the difference in scientific notation. 58.7 * 10^5 ⇔ 5.87 * 10^6 The difference in produced energies of the two type of wind turbines is 5.87 * 10^6 kilowatt-hours per year.
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Pop Quiz
Add or Subtract Numbers in Scientific Notation
Perform the appropriate operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.
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Closure
Writing the Distance From the Earth to the Sun
The distance from the Earth to the Sun is about 150 000 000 kilometers.
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Scientific Notation
Exercise 2.1
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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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Hint & Answer
S
Solution
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.
What is the combined mass of Earth and the Moon expressed in scientific notation?
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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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