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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.**

Challenge

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?

Discussion

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 $10_{2}=100.$

Example

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 $

Example

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 $

Pop Quiz

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

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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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"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"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Turkey:","formTextAfter":"people","answer":{"text":["8 \\times 10^7"]}}

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Australia:","formTextAfter":"people","answer":{"text":["3 \\times 10^7"]}}

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} $

Discussion

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.

Discussion

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} $

Example

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} $

Example

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"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"About","formTextAfter":"hours","answer":{"text":["2.08 \\times 10^{-2}"]}}

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 $10_{1}.$