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Scientists and astronomers make calculations using huge and tiny numbers. They could go all the way up to $100000000000$ or as low as $0.00000000001.$ 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.**

The distance from the Earth to the Sun is about $150000000$ 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?

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.

### 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 __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.
__left to right__ to make the number greater than or equal to $1$ and less than $10.$ In this case, the number of places moved indicates the negative exponent to be used for the base $10$ power.

$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.
$4000000=4×1000000=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÷10000=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 |
---|---|---|

$4505$ | $4.505×1000$ | $4.505×10_{3}$ |

$8320000$ | $8.32×1000000$ | $8.32×10_{6}$ |

$0.0005$ | $5÷10000$ | $5×10_{-4}$ |

$0.0521$ | $5.21÷100$ | $5.21×10_{-2}$ |

move. Consider a number greater than $10.$ The decimal would move from

Similarly, for numbers less than $1,$ such as $0.000022,$ the decimal will move from

Scientific notation is not only a convenient way to express cumbersome numbers. It also eases the comparison of numerical order of magnitude. For example, it may be difficult to determine how much larger $237000000$ is compared to $4530000.$ However, it is easier to see that $2.37×10_{8}$ and $4.5×10_{6}$ differ by a factor of about $10_{2}=100.$

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 $8848$ 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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a The height of Mount Everest is $8848$ 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 Form8848 Scientific Notation8.848×10_{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 Notation1.851×10_{4} Standard Form18510 $

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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a Start by placing the decimal point after the first non-zero digit.

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.

$0.000002.↑First nonzero digit $

Then, determine the power of $10.$ That is done by counting the number of digits $0.000002 .↑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 Form0.000002 Scientific Notation2×10_{-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 Notation1.5×10_{-6} Standard Form0.0000015 $

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.

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.
### Solution

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.

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 $10_{7}.$

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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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 | $331002651$ | $300000000$ |

Brazil | $212559417$ | $200000000$ |

Turkey | $84339067$ | $80000000$ |

China | $1439323776$ | $1000000000$ |

Australia | $25499884$ | $30000000$ |

$300000000=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 | $331002651$ | $300000000$ | $3×10_{8}$ |

Brazil | $212559417$ | $200000000$ | $2×10_{8}$ |

Turkey | $84339067$ | $80000000$ | $8×10_{7}$ |

China | $1439323776$ | $1000000000$ | $1×10_{9}$ |

Australia | $25499884$ | $30000000$ | $3×10_{7}$ |

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 | $3×10_{7}$ |

$USA3×10_{8} Brazil2×10_{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.
$Turkey8×10_{7} Australia3×10_{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.
$ChinaUSABrazilTurkeyAustralia 1×10_{9}3×10_{8}2×10_{8}8×10_{7}3×10_{7} $

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.

$(a×10_{b})×(c×10_{d})=ac×10_{b+d}$

$(1.5×10_{2})×(12×10_{5}) $

Three steps can be followed to multiply these numbers.
1

Write Each Number in Scientific Notation

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 $10.$

$First Factor1.512 ××× Second Factor10_{2}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

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

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

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.

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.

$c×10_{d}a×10_{b} =ca ×10_{b−d}$

$12000.36×10_{23} $

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

Start by checking whether each number is written in scientific notation.

$First Factor0.36 ×× Second Factor10_{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

Now that both the dividend and divisor are written in scientific notation, the first factors of both can be divided.

$1.2×10_{3}3.6×10_{22} $

WriteProdFrac

Write as a product of fractions

$1.23.6 ×10_{3}10_{22} $

CalcQuot

Calculate quotient

$3×10_{3}10_{22} $

3

Divide the Second Factors

4

Write the Result in Scientific Notation

Finally, write the resulting number in scientific notation. Notice that in this instance, the quotient is already in scientific notation.

$3×10_{19} $

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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a Use the Commutative Property of Multiplication and the Product of Powers Property.

b Use the Quotient of Powers Property.

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 $10.$

$First Factor1.45 ×× Second Factor10_{44}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

$a_{m}⋅a_{n}=a_{m+n}$

$7×10_{58}$

$First Factor1.45 ×× Second Factor10_{44}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.
$5×10_{14}1.4×10_{44} $

WriteProdFrac

Write as a product of fractions

$51.4 ×10_{14}10_{44} $

CalcQuot

Calculate quotient

$0.28×10_{14}10_{44} $

DivPow

$a_{n}a_{m} =a_{m−n}$

$0.28×10_{30}$

$0.28×10_{30}⇔2.8×10_{29} $

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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a Distance traveled can be calculated by multiplying the speed by the time.

b Use the Quotient of Powers Property.

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

$a_{m}⋅a_{n}=a_{m+n}$

$11.52×10_{-1}$

$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.

$1meter=0.001kilometers $

Now, rearrange the distance formula to use it for the $Distance=Speed×Time⇓Time=SpeedDistance $

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.
$4.8×10_{-2}1×10_{-3} $

WriteAsFrac

Write as a fraction

$4.81 ×10_{-2}10_{-3} $

UseCalc

Use a calculator

$(0.208333…)×10_{-2}10_{-3} $

RoundDec

Round to $3$ decimal place(s)

$0.208×10_{-2}10_{-3} $

DivPow

$a_{n}a_{m} =a_{m−n}$

$0.208×10_{-3−(-2)}$

SubNeg

$a−(-b)=a+b$

$0.208×10_{-3+2}$

AddTerms

Add terms

$0.208×10_{-1}$

$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!
Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.

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.

$(a×10_{b})±(c×10_{b})=(a±c)×10_{b}$

$(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

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

The obtained numbers are now like terms. That means the first factors are ready to be added.

3

Write the Result in Scientific Notation

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} $

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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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.

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.
$(6×10_{6})(60×10_{5}) −⇓− (1.3×10_{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

$58.7×10_{5}$

$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.
The initial challenge of this collection stated that the distance from the Earth to the Sun is about $150000000$ kilometers. ### Answer

### Solution

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?

Yes, by using scientific notation.

This number can be written as a product of two numbers to express it in shorter notation.
*scientific notation*. A number written in scientific notation needs to have the following two certain characteristics.

$150000000=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.
$150000000⇓15×10_{7} $

Consider the main topic of this lesson, - The first factor must be greater than or equal to $1$ and less than $10.$
- The second factor must be a power of $10.$

$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 $150000000$ 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!