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

## Distance From the Earth to the Sun

The distance from the Earth to the Sun is about 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

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.
In this form, the first factor is greater than or equal to and less than In other words, it needs to be in the interval The second factor is a power of where is an integer. For example, the number million can be rewritten as the product of and a multiple of Then, the multiple of is rewritten as a base power.
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 as an example.
In such cases, numbers are expressed as a division by a multiple of Division by a multiple of is equivalent to multiplication by a base 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

### 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 The decimal would move from right to left to make the number less than but still greater than The number of places the decimal moved indicates the positive exponent to be used for the base power.
Similarly, for numbers less than such as the decimal will move from left to right to make the number greater than or equal to and less than In this case, the number of places moved indicates the negative exponent to be used for the base power.
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 is compared to However, it is easier to see that and differ by a factor of about
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 meters above the sea level. Rewrite this height in scientific notation.
b Mount Elbrus is an extinct volcano with twin cones that reach feet. Rewrite this height in standard form.

### Hint

a Start by placing the decimal point after the first non-zero digit.
b The power of is positive. That means the decimal point moves to the right.

### Solution

a The height of Mount Everest is 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.
Next, count the digits after the decimal point.
There are digits after the decimal point. This number will be written as the exponent of
b This time the number is given in the scientific notation and the task is to rewrite it into standard form.
The power of is It has a positive exponent. Therefore, the decimal point will be moved to the right times. In other words, the first factor will be multiplied by 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.
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.
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 meters. Write this length in scientific notation in meters.
b The length of a salmonella bacterium is meters. Rewrite this length in standard form.

### Hint

a Start by placing the decimal point after the first non-zero digit.
b The power of is negative. That means the decimal point moves to the left.

### Solution

a The length of an Escherichia coli bacterium is given as meters. The goal is to write this number in scientific notation. Begin by placing the decimal point after the first non-zero digit.
Then, determine the power of That is done by counting the number of digits before the new decimal point.
There are digits before the decimal point. The given number is less than That means the exponent will be negative.
b The length of a salmonella bacterium is given in scientific notation.
Notice that the power of is negative. This means that the decimal point will be moved to the 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.
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.

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.
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.
b Sort the countries from greatest to least population.

### Hint

a Round the numbers to the greatest place value. Then count the zeros.
b Examine the powers of for each country. Then, compare the first factors of the numbers with the same power of

### 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
Brazil
Turkey
China
Australia
Now all the rounded numbers can be rewritten as a single digit times a power of Count the zeros to determine the power of for each number. For instance, the rounded population of the USA can be rewritten in this way.
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
Brazil
Turkey
China
Australia
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
Brazil
Turkey
China
Australia
Examine the powers of 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 Start with because is greater than
The powers of for the US and Brazil are equal. That means their first factors should be compared. The value is larger than That means the population of the USA is greater than Brazil. Next, compare the populations of Turkey and Australia.
Again, the powers of are the same. This indicates that the values of and should be checked. Well, is greater than That means the population of Turkey is greater than Australia. Now the countries can be sorted from greatest to least population.
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 and can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.

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 the Product of Powers Property can be used. As an example, consider the following product.
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 The other number should be greater than or equal to and less than
The first number is in scientific notation. The second number is not. Its first factor is greater than Note that can also be expressed as The decimal point needs to move one unit to the left for the first factor to become less than The second factor's power of is then increased by that number of moves,
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.
3
Multiply the Second Factors
expand_more
Next, the powers with base 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 The first factor is greater than and less than The second factor is written as a power of Therefore, the result is already in scientific notation.

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.

Consider the division of the following two numbers.
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.
The first factor is less than That means the dividend still needs to be written in scientific notation. Move the decimal point to the right until is greater than and less than Then, decrease the second factor's exponent by the number of moves.
Next, consider the divisor. The divisor is a whole number. Note that can also be expressed as Now, move the decimal point three units to the left to get a first factor that is less than and greater than The second factor can then be written as to the power of
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
Divide the Second Factors
expand_more
Next, use the Quotient of Powers Property to divide the second factors. This property states that when dividing powers with the same base, the exponents are subtracted.
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.
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.
b Solve along with Ramsha to find the quotient given in Example II. Express the result in scientific notation.

### 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.
1. The first factor must be greater than or equal to and less than
2. The second factor must be a power of
It is helpful to organize this check in the following way.
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.
The product of the multiplication is already in scientific notation!
b Notice that Example II uses the same numbers as Example I. It was found in Part A that both the dividend and divisor are already written in scientific notation.
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.
Notice that the result is not in scientific notation yet because the first factor is less than The decimal point needs to be moved one unit to the right. That move means the power of must be decreased by one. Now it is in scientific notation.
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 kilometers per hour!

a How many kilometers can a snail go if it is on the move for a full day of hours? Perform the needed operations in scientific notation. Also express the result in scientific notation. Round the result to two decimal places if necessary.
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.

### Hint

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

### Solution

a It is given that a snail can move kilometers in one hour. One day is hours, so the distance traveled in one day can be found by multiplying the speed of the snail by
Begin by rewriting both numbers in scientific notation. Notice that is less than The decimal point moves two units to the right to become greater than Next, the two unit move to the right means the base power will have an exponent of .
Now consider A number greater than 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 power will be positive. The second factor is