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. 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 to move
. 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.
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Example
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
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Example
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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Pop Quiz
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.
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Example
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.
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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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"Brazil:","formTextAfter":"people","answer":{"text":["2 \\times 10^8"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"Turkey:","formTextAfter":"people","answer":{"text":["8 \\times 10^7"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"China:","formTextAfter":"people","answer":{"text":["1 \\times 10^9"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"Australia:","formTextAfter":"people","answer":{"text":["3 \\times 10^7"]}}
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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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)
The first factors of the numbers are multiplied like integers or decimal numbers. Then, the exponents of the second factors are added. Since they have the same base 10, the Product of Powers Property can be used. As an example, consider the following product. (1.5 * 10^2)*(12 * 10^5) Three steps can be followed 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. 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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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 These numbers written in scientific notation, and they can be divided in four steps. It is similar to the process of multiplying numbers written in scientific notation.
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)
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Example
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.
It is helpful to organize this check in the following way. c First Factor & &Second Factor 1.4 & * & 10^(44) & ✓ 5 & * & 10^(14) & ✓ Both numbers meet the two characteristics. They are written in scientific notation. Now their first factors can be multiplied by using the Commutative Property of Multiplication. Their second factors can be multiplied 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!
c First Factor & &Second Factor 1.4 & * & 10^(44) & ✓ 5 & * & 10^(14) & ✓ Recall how to divide numbers written in scientific notation. The first factors are divided like fractions and the second factors are divided 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)
Notice that the result is not in scientific notation yet because the first factor is less than 1. The decimal point needs to be moved one unit to the right. That move means the power of 10 must be decreased by one. Now it is in scientific notation. 0.28 * 10^(30) ⇔ 2.8 * 10^(29)
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Example
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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"About","formTextAfter":"hours","answer":{"text":["2.08 \\times 10^{-2}"]}}
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.
Distance = Speed * Time ⇓ Distance= 0.048 * 24 Begin by rewriting both numbers in scientific notation. Notice that 0.048 is less than 1. The decimal point moves two units to the right to become greater than 1. Next, the two unit move to the right means the base 10 power will have an exponent of -2. 0.048 * 10^0 = 4.8 * 10^(-2) Now consider 24. A number greater than 10 moves a certain number of units to the left. This case requires a one unit move to the left. Recall that 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, 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)
The result is not in scientific notation. It will need to be rewritten. Follow the same method as done previously. Move the decimal point one unit to the left and increase the power of 10 by that same value, 1. 11.52 * 10^(-1) = 1.152 * 10^0 Finally, round the obtained 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. Quite impressive!
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 meter = 0. 001 kilometers Now, rearrange the distance formula to use it for the time by dividing both sides of the equation by speed. Distance = Speed * Time ⇓ Time=Distance/Speed Before using the obtained formula, rewrite the numbers to have them in scientific notation. To do so, move the decimal points to the right to make the first factors greater than or equal to 1 and decrease the powers of 10 according to amount of decimal points moved. 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)
Since the first factor of the obtained result is less than 1, move the decimal point one unit to the right and decrease the power of 10 one unit as well to have it 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. For those curious, 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 the powers of 10 are the same.
( a* 10^b) ± ( c* 10^b)=( a ± c) * 10^b
Recall that rewriting the result in scientific notation is necessary when the first factor is greater than 10 or less than 1. In such cases, the exponent of 10 is increased or decreased by moving the decimal point. Consider the following addition example. (3 * 10^(12))+(0.15 * 10^(15)) Adding these numbers calls for three steps to be followed.
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 * 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 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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Example
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.
(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 one unit to the right. That results in its power of 10 decreasing by one unit. 6 * 10^6 ⇔ 60 * 10^5 The first factors of the numbers can be added since these numbers are now like terms.
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 can be moved one unit to the left. That results in its power of 10 increasing by one. 61.3 * 10^5 ⇔ 6.13 * 10^6 In total, a typical large and small wind turbine produce 6.13 * 10^6 kilowatt-hour energy per year. That is enough to meet the electricity demand of around 1600 average households per year.
b
This time the difference between the energy amounts will be found.
(6 * 10^6) - (1.3 * 10^5) Perform this operation by setting the powers of 10 as the same. Recall that the first number was already rewritten in Part A. ccc ( 6 * 10^6) & - & (1.3 * 10^5) & ⇓ & ( 60 * 10^5) & - & (1.3 * 10^5) Subtraction can be performed now that they are like terms.
(60 * 10^5) - (1.3 * 10^5)
FactorOut
Factor out 10^5
(60-1.3) * 10^5
SubTerms
Subtract terms
58.7 * 10^5
The result is not in scientific notation. It can be rewritten by moving the decimal point one unit to the left. That results in the power of 10 increasing by one unit. 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-hour per year.
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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.
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!
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Scientific Notation
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We are given a number and we want to determine whether it is written in scientific notation. 32 * 10^7 Recall that a number in scientific notation is the product of two factors. The first factor has to be a number greater than or equal to 1 and less than 10. The second factor is a power of 10. Notice that our number is given as the product of two factors.
| 32 * 10^7 | |
|---|---|
| First Factor | Second Factor |
| 32 | 10^7 |
The first factor 32 is greater than 10. That does not meet the conditions of scientific notation. That means the given number is not written in scientific notation.
Once again, we will examine the factors of the given product to determine whether it is written in scientific notation or not. 5.6 * 10^(-3) The first factor has to be a number greater than or equal to 1 and less than 10. The second factor is a power of 10.
| 5.6 * 10^(-3) | |
|---|---|
| First Factor | Second Factor |
| 5.6 | 10^(-3) |
Notice that 5.6 is less than 10 and greater than 1. Also the second factor 10^(-3) is a power of 10. These characteristics meet the conditions of scientific notation. That means the given number is written in scientific notation.
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A number written in scientific notation usually expresses a very large or very small number. It does so by writing its value as a product of 10 to some power. Let's look at some examples.
| Standard Notation | Scientific Notation |
|---|---|
| 5 0 000 000 000 | 5 * 10^(10) |
| 5 00 000 | 5 * 10^5 |
| 5 0 | 5 * 10^1 |
| 5 | 5 * 10^0 |
| 0.5 | 5 * 10^(- 1) |
| 0.00005 | 5 * 10^(- 5) |
| 0.0000000005 | 5 * 10^(- 10) |
Consider the steps needed to change from standard form to scientific notation. First, we need to move the decimal point until the resulting number is greater than 1 and less than 10. Then, the number of places the decimal moves will be the exponent of 10.
Notice that we placed the decimal point after the first non-zero digit. There are 11 digits after the decimal point. Therefore, the scientific notation of 631 200 000 000 is 6.312* 10^(11). Standard Form:& 631 200 000 000 Scientific Notation:& 6.312* 10^(11)
This time we are given a number less than 1. We want to go from standard form to scientific notation. Once again, we will move the decimal point until the resulting number is greater than 1 and less than 10. The number of places the decimal moves will be the exponent of 10.
We count the number of digits before the decimal point. Therefore, the exponent of 10 will be negative. Since we moved the decimal point 8 digits to the right, the scientific notation of 0.000000055 is 5.5* 10^(- 8). Standard Form:& 0.000000055 Scientific Notation:& 5.5* 10^(- 8)
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Notice that 7 * 10^(- 6) is a number written in scientific notation. This notation usually expresses very large or very small numbers written as a product of 10 to some power. We are trying go from scientific notation to standard form. Do that by moving the decimal point of the given number according to the exponent of 10. 7* 10^(- 6) In this case, the exponent is - 6. Since the exponent is a negative number, we will move the decimal point six places to the left.
Therefore, the standard form of 7* 10^(- 6) is 0.000007. Scientific Notation:& 7* 10^(- 6) Standard Form:& 0.000007
Once again, we have a number 4.2 * 10^8 written in scientific notation. Let's have a look at the exponent of 10. 4.2* 10^8 In this case, the exponent is 8. The exponent is a positive number. That means we will move the decimal point eight places to the right.
The standard form of 4.2* 10^8 is 420 000 000. Scientific Notation:& 4.2* 10^8 Standard Form:& 420 000 000
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E. The number that follows
Erepresents the exponent of 10. Maya used a calculator to evaluate some large and small numbers. She ended with the following numbers on the screen. Write these numbers in standard form.
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The number displayed on a calculator screen is written in scientific notation. Note that 9.8 E 11 means that 9.8 is multiplied by 10^(11). 9.8 E 11 ⇔ 9.8 * 10^(11) We will move the decimal of the given number according to the exponent of 10 to change from scientific notation to standard form. In this case, the exponent is positive11. That means we should move the decimal point eleven places to the right.
The standard form of 9.8* 10^(11) is 980 000 000 000. Scientific Notation:& 9.8* 10^(11) Standard Form:& 980 000 000 000
Once again, we will first write the number displayed on the calculator screen in scientific notation. Recall that the number following E
represents the power of 10.
6 E -10 ⇔ 6 * 10^(-10)
The exponent is negative10. That means we move the decimal point ten places to the left.
The standard form of 6* 10^(- 10) is 0.0000000006. Scientific Notation:& 6* 10^(- 10) Standard Form:& 0.0000000006
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Maya loves reading science magazines. Recently, she saw the following table which shows the masses of the planets.
| Name of the Planet | Mass of the Planet (kg) |
|---|---|
| Mercury | 3.30 * 10^(23) |
| Venus | 4.87 * 10^(24) |
| Earth | 5.97 * 10^(24) |
| Mars | 6.42 * 10^(23) |
| Jupiter | 1.90 * 10^(27) |
| Saturn | 5.68 * 10^(26) |
| Uranus | 8.68 * 10^(25) |
| Neptune | 1.02 * 10^(26) |
Order the planets from the greatest mass to the smallest mass.
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Notice that the masses of the planets are already given in scientific notation. In other words, the first factors are greater than 1 and less than 10. The second factors are the power of 10. We will first have a look at the powers of 10 to compare these numbers.
| Name of the Planet | Mass of the Planet (kg) |
|---|---|
| Mercury | 3.30 * 10^(23) |
| Venus | 4.87 * 10^(24) |
| Earth | 5.97 * 10^(24) |
| Mars | 6.42 * 10^(23) |
| Jupiter | 1.90 * 10^(27) |
| Saturn | 5.68 * 10^(26) |
| Uranus | 8.68 * 10^(25) |
| Neptune | 1.02 * 10^(26) |
Two things are happening. Some numbers have different powers of 10 and some numbers have the same power of 10. If the bases are the same, then the number with the greater power is greater. Let's sort the powers of 10 with that in mind. 10^(23) < 10^(24) < 10^(25) < 10^(26) < 10^(27) Note that the greatest power of 10 is 10^(27). Only one number has this factor. That means Jupiter — mass of 1.90 * 10^(27) kilograms — is the planet with the greatest mass. Let's now examine the numbers with the factor 10^(26). Recall that we only need to compare the first factors if the second factors are the same. 1.02 * 10^(26) < & 5.68 * 10^(26) Neptune < & Saturn The mass of Neptune is less than the mass of Saturn. That is because 1.02 is less than 5.68. Let's continue with the numbers with the second factor 10^(25). The mass of Uranus 8.68 * 10^(25) is the only planet with this factor. We can conclude that it is the planet with a mass less than Neptune's. Let's order the planets we analyzed so far! Start with the greatest mass. Jupiter & 1.90 * 10^(27) Saturn & 5.68 * 10^(26) Neptune & 1.02 * 10^(26) Uranus & 8.68 * 10^(25) Next, we will examine the first factors of the numbers including 10^(24) as a second factor. 4.87 * 10^(24) < & 5.97 * 10^(24) Venus < & Earth Since 5.97 is greater than 4.87, the mass of Earth is more than Venus. Let's see the order with these two planets included. The top of the list goes from the planet with the greatest mass to the planet with the smallest mass. Jupiter & 1.90 * 10^(27) Saturn & 5.68 * 10^(26) Neptune & 1.02 * 10^(26) Uranus & 8.68 * 10^(25) Earth & 5.97 * 10^(24) Venus & 4.87 * 10^(24) Last, we will compare the numbers with the second factor 10^(23). 3.30 * 10^(23) < & 6.42 * 10^(23) Mercury < & Mars The mass of Mars is more than the mass of Mercury. Finally, we can list the masses in order. Let's show it in a table!
| Name of the Planet | Mass of the Planet (kg) |
|---|---|
| Jupiter | 1.90 * 10^(27) |
| Saturn | 5.68 * 10^(26) |
| Neptune | 1.02 * 10^(26) |
| Uranus | 8.68 * 10^(25) |
| Earth | 5.97 * 10^(24) |
| Venus | 4.87 * 10^(24) |
| Mars | 6.42 * 10^(23) |
| Mercury | 3.30 * 10^(23) |
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